SF 


GIFT   OF 
MICHAEL  REE^E 


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WORKS  OF 
PROF.  HENRY  T.  BOVEY 


PUBLISHED    BY 


JOHN  WILEY  &  SONS. 


A  Treatise  on  Hydraulics. 

8vo,  cloth,  $4.00. 

Strength  of  Materials  and  Theory  of  Structures. 

830  pages,  8vo,  cloth,  $7.50. 


A   TREATISE 


ON 


HYDRAU  LICS. 


BY 

HENRY    T.    ^JO 

M.  INST.C.E.,  LL.D.,  F.R.S.C., 

Professor  of  Civil  Engineering  and  Applied  Mechanics, 
McGill  University,  Montreal. 


FIRST  EDITION. 

FIRST  THOUSAND. 


NEW  YORK: 

JOHN  WILEY  &  SONS. 

LONDON  :  CHAPMAN  &  HALL,  LIMITED, 

1895. 


Copyright,  1895, 

BY 

HENRY  T.  BOVEY. 


ROBERT   DRUMMOND,    HLHCTROTYPBR   AND   PRINTER,   NEW  YORK. 


PREFACE. 


THE  present  treatise  is  the  outcome  of  lectures  delivered  in 
McGill  University  during  the  last  ten  or  twelve  years,  and 
although  intended  primarily  for  the  use  and  convenience  of 
the  student  of  hydraulics,  it  is  hoped  that  it  may  also  prove 
acceptable  to  the  engineer  in  general  practice. 

In  order  to  render  the  treatment  of  the  subject  more  com- 
plete, free  reference  has  been  made  to  standard  authors  on  the 
subject.  The  examples  introduced  to  illustrate  the  text  have 
also  been  selected  in  part  from  the  works  of  such  well-known 
writers  as  Weisbach,  Osborne  Reynolds,  and  Cotterill,  but  the 
greater  number  are  such  as  have  occurred  in  the  course  of  the 
author's  own  experience.  The  tables  of  coefficients  of  discharge 
have  been  prepared  from  the  results  of  experiments  carried 
out  in  the  Hydraulic  Laboratory  of  the  University.  These 
experiments  are  still  being  continued  and  may  probably  form 
the  subject  of  a  special  paper. 

The  author  desires  to  acknowledge  many  suggestions 
offered  by  Professor  Bamford,  and  to  express  his  deep  obliga- 
tion to  Professor  Chandler  for  much  labor  and  time  given  to 
the  revision  of  proof  sheets. 

HENRY  T.  BOVEY. 

MONTREAL,  November,  1895. 


CONTENTS. 


CHAPTER    I. 

FLOW   THROUGH    ORIFICES   AND    OVER   WEIRS. 

PAGE 

Definitions i 

Stream-line  Motion 2 

Motion  in  Plain  Layers 2 

Laminar  Motion 2 

Density  of  Water 2 

Continuity 2 

Bernouilli's  Theorem 6 

Applications  of  Bernouilli's  Theorem 9 

Piezometer 9 

Orifice  in  a  Thin  Plate 13 

Torricelli's  Theorem 14 

Efflux  from  Orifice  in  a  Vessel  in  Motion 16 

Flow  in  a  Frictionless  Pipe  of  Gradually  Changing  Section 18 

Hydraulic  Resistances 20 

Coefficient  of  Velocity 20 

Coefficient  of  Resistance 21 

Coefficient  of  Contraction 22 

Coefficient  of  Discharge 24 

Miner's  Inch 26 

Energy  and  Momentum  of  Jet 27 

Inversion  of  the  Jet 27 

Time  Required  to  Empty  and  Fill  a  Lock 29 

General  Equations 30 

Loss  of  Energy  in  Shock 32 

Mouthpieces 34 

Borda's  Mouthpiece 34 

Ring  Nozzle 37 

Cylindrical  Mouthpiece 39 

Divergent  Mouthpiece 42 

Convergent  Mouthpiece 44 

Radiating  Current 46 

Vortex  Motion 47 

Free  Spiral  Vortex 48 

Forced  Vortex 49 

v 


vi  CONTENTS. 


Compound  Vortex 50 

Large  Orifices 50 

Rectangular  Orifices  of  Large  Size 50 

Circular  Orifices  of  Large  Size 53 

Notches 54 

Weirs 54 

Triangular  Notch 56 

Broad-crested  Weir 58 

Examples 60 

CHAPTER    II. 

FLUID  FRICTION. 

Definition 70 

Laws  of  Fluid  Friction 72 

Surface  Friction  in  Pipes 73 

Resistance  of  Ships 76 

CHAPTER   III. 

FLOW  IN  PIPES. 

Assumptions .* 78 

Steady  Motion .. 78 

Influence  upon  the  Flow  of  the  Pipe's  Position 83 

Transmission  of  Energy  by  Hydraulic  Pressure 84 

Flow  in  a  Uniform  Pipe  Connecting  Two  Reservoirs 86 

Losses  of  Head  due  to  Abrupt  Changes  of  Section 89 

Remarks  on  the  Law  of  Resistance  to  Flow 96 

Flow  in  a  Pipe  of  Varying  Diameter 98 

Equivalent  Uniform  Main 100 

Branch  Main  of  Uniform  Diameter 101 

Nozzles 104 

Motor  Driven  by  Hydraulic  Pressure 107 

Siphons 108 

Inverted  Siphons 109 

Air  in  Pipe no 

Three  Reservoirs  Connected  by  a  Branched  Pipe m 

Orifice  Fed  by  Two  Reservoirs 115 

Variation  of  Velocity  in  a  Transverse  Section 119 

Examples 122 

CHAPTER   IV. 

FLOW  OF  WATER  IN  OPEN  CHANNELS. 

Flow  of  Water  in  Channels t 131 

Steady  Flow  in  Channels  of  Constant  Section 132 


CONTENTS.  Vll 


Form  of  a  Channel 135 

Flow  in  Aqueducts 142 

River  Bends 143 

Value  of/ 144 

Darcy  and  Bazin's  Formulae 145 

Ganguillet  and  Kiitter's  Formulas 147 

Variation  of  Velocity  in  a  Transverse  Section 148 

Bazin's  Formula 152 

Boileau's  Formula 153 

Relations  between  Surface,  Mean,  and  Bottom  Velocities 154 

Flow  of  Water  in  Open  Channels  of  Varying  Cross-section 156 

Standing  Wave 165 

Examples 170 

CHAPTER   V. 
METHODS  OF  GAUGING. 

Gauging  of  Streams  and  Watercourses 173 

Hook  Gauge '. 173 

Surface-floats 175 

Subsurface-floats 175 

Twin-floats 176 

Velocity  Rod 176 

Pitot  Tube 176 

Darcy  Gauge 178 

Current  Meters 180 

Hydrometric  Pendulum 183 

Gauging  of  Pipes 183 

Venturi  Meter 183 

Piston  Meter 184 

Inferential  Meter 184 

CHAPTER   VI. 
IMPACT. 

Impact  upon  a  Flat  Vane  Oblique  to  Direction  of  Jet 186 

Impact  upon  a  Flat  Vane  Normal  to  Direction  of  Jet 189 

Reaction 190 

Jet  Propeller 190 

Impact  upon  a  Surface  of  Revolution 192 

Impact  upon  a  Flat  Vane  with  Rim 195 

Pressure  in  a  Pipe  upon  a  Thin  Plate  Normal  to  the  Direction  of  Motion.  196 
Pressure  in  a  Pipe  upon  a  Cylindrical  Body  about  Three  Diameters  in 

Length 198 

Impact  upon  a  Curved  Vane 199 

Frictional  Effect 205 

Resistance  to  the  Motion  of  a  Solid  in  a  Fluid  Mass 205 

Examples ; 209 


Vlll  CONTENTS. 

CHAPTER   VII. 
HYDRAULIC  MOTORS  AND  CENTRIFUGAL  PUMPS. 

PAGE 

Classification 213 

Hydraulic  Ram 214 

Pressure  Engine 215 

Accumulator 215 

Losses  of  Energy  in  Pressure  Engines 221 

Hydraulic  Brakes 223 

Water-wheels ^ 225 

Undershot  Wheels 225 

Wheels  in  a  Straight  Race 227 

Poncelet  Wheel 232 

Form  of  Bucket 240 

Breast-wheels 242 

Sluices 244 

Overshot  Wheels 254 

Effect  of  Centrifugal  Force 255 

Weight  of  Water  on  Wheel 256 

Arc  of  Discharge 256 

Pitch-back  Wheel 272 

Ventilated  Bucket 272 

Jet  Reaction  Wheel •> 272 

Barker's  Mill 272 

Scotch  Turbine 276 

Reaction  Turbines 276- 

Impulse  Turbines 276 

Hurdy-gurdy  Wheel 279 

Pelton  Wheel 280 

Radial-,  Axial-,  and  Mixed-flow  Turbines 281-284 

Limit  Turbine 283 

Theory  of  Turbine     284 

Remarks  on  Centrifugal  Head  in  Turbine-flow 298 

Practical  Values  of  the  Velocities,  etc. ,  in  Turbines 299 

Theory  of  the  Section-tube 301 

Losses  of  Energy  in  Turbines 303 

Centrifugal  Pumps ." 307 

Theory  of  Centrifugal  Pumps 309 

Examples 315 


HYDRAULICS. 


CHAPTER   I. 
FLOW  THROUGH   ORIFICES,  OVER  WEIRS,   ETC. 

I.  Fluid  Motion. — The  term  "  hydraulics,"  as  its  derivation 
(vdoop,  water ;  avXos,  a  tube  or  pipe)  indicates,  was  primarily 
applied  to  the  conveyance  of  water  in  a  tube  or  pipe,  but  its 
meaning  now  embraces  the  experimental  theory  of  the  motion 
of  fluids. 

The  motion  of  a  fluid  is  said  to  be  steady  or  permanent 
when  the  molecules  successively  arriving  at  any  given  point 
are  animated  with  the  same  velocity,  are  subjected  to  the 
same  pressure,  and  are  the  same  in  density.  As  soon  as  the 
motion  of  a  stream  becomes  steady  a  permanent  regime  is  said 
to  be  established,  and  hydraulic  investigations  are  usually 
made  on  the  hypothesis  of  a  permanent  regime.  With  such 
an  hypothesis  any  portion  of  the  fluid  mass  which  leaves  a 
given  region  is  replaced  by  a  like  portion  under  conditions 
which  are  identically  the  same. 

The  terms  "steady  motion"  and  " permanent  regime"  are 
often  considered  to  be  synonymous. 

The  general  problem  of  flow  is  the  determination  of  the 
relation  which  exists  at  any  point  between  the  density,  press- 


2  HYDRA  ULICS. 

ure,  and  velocity  of  the  molecules  which  successively  pass  that 
point. 

The  actual  motion  of  a  fluid  is  exceedingly  complex,  and 
in  order  to  simplify  the  investigations  various  assumptions  are 
made  as  to  the  nature  of  the  flow. 

2.  (a)  Stream-line  Motion. — The    molecules   may  be    re- 
garded as  flowing  along  definite  paths,  and  a  succession  of  such 
molecules  will  form  a  continuous  fluid  rope  which  is  termed  an 
elementary  stream  or  a  fluid  filament,  or,  if  the  motion  is  steady 
and  the  paths  therefore  fixed  in  space,  a  stream-line. 

Experiment  shows  that  the  velocity  of  flow  in  any  cross- 
section  varies  from  point  to  point,  and  hence  it  is  often  assumed 
that  the  section  is  made  up  of  an  infinite  number  of  indefi- 
nitely small  areas,  each  area  being  the  section  of  a  fluid 
filament. 

(b)  Motion  in  Plane  Layers. — In  this  motion  it  is  assumed 
that  the  molecules  which  at  any  given  moment  are  found  in  a 
plane  layer  will  remain  in  a  plane  layer  after  they  have  moved 
into  any  new  position. 

(c)  Laminar  Motion.— On   this   hypothesis  the  stream  is 
supposed  to  consist  of  an  infinite  number  of  indefinitely  thin 
layers.     The  variation   in   velocity  from   point  to   point   of  a 
cross-section  may  then  be  allowed  for  by  giving  the  several 
layers  different  velocities  based  upon  the  law  of  fluid  resistance 
between  consecutive  layers. 

3.  Density;  Compressibility;  Head;  Continuity. 

The  weight  of  ice  per  cubic   foot  at  23°  F.    is  57.2  Ibs.; 

"freshwater"        "         "     "  39.2°  F.  is  62.425  Ibs.; 
"        "        "salt       "       "        "        "     "  53°  F.    is  64  Ibs.; 
"fresh     "       "       "         "     "  53°  F.     is  62.4  Ibs., 

or  1000  kilog.  per  cubic  metre. 

The  following  table  from  the  article  on  "  Hydromechanics  " 
in  the  Encyc.  Brit,  gives  the  density  of  water  at  different 
temperatures: 


FLOW   THROUGH  ORIFICES,   OVER    WEIRS,  ETC. 


Temperature. 

Density. 

Weight 
in  Lbs.  per 
Cu.  Ft. 

Temperature. 

Density. 

Weight 
in  Lbs.  per 
Cu.  Ft. 

Cent. 

Fahr. 

Cent. 

Fahr. 

0 

32 

.999884 

62.417 

20 

68 

.998272 

62.316 

I 

33-8 

.999941 

62.420 

22 

71-6 

.997839 

62.289 

2 

35-6 

.999982 

62.423 

24 

75-2 

.997380 

62.261 

3 

37-4 

I  .  000004 

62.424 

26 

78.8 

.996879 

62.229 

4 

39-2 

1.000013 

62.425 

28 

82.4 

.996344 

62.196 

5 

4i 

1.000003 

62.424 

30 

86 

•995778 

62.  161 

6 

42.8 

.999983 

62.423 

35 

95 

.994690 

62.093 

7 

44.6 

•999946 

62.421 

40 

104 

.992360 

61.947 

8 

46.4 

.999899 

62.418 

45 

H3 

.990380 

61.823 

9 

48.2 

•999837 

62.414 

50 

122 

.988210 

61.688 

10 

50 

.999760 

62.409 

55 

131 

.985830 

61.540 

ii 

51.8 

.999668 

62.403 

60 

140 

.983390 

61.387 

12 

53-6 

.999562 

62.397 

65 

I49 

.980750 

6l.222 

13 

55-4 

.999443 

62.389 

70 

158 

•977950 

61.048 

14 

57-2 

.999312 

62.381 

75 

167 

.974990 

60.863 

15 

59 

•999!73 

62.373 

80 

176 

.971950 

60.674 

16 

60.8 

.999015 

62.363 

85 

185 

.968800 

60.477 

17 

62.6 

.998854 

62.353 

90 

194 

.965570 

60.275 

18 

64.4 

.998667 

62.341 

IOO 

212 

.958660 

59.844 

19 

66.2 

.998473 

62.329 

Fluids  are  sensibly  compressed  under  heavy  pressures,  and 
the  compression  is  proportional  to  the  pressure  up  to  about 
1000  Ibs.  (65  atmospheres)  per  square  inch.  Grassi's  ex- 
periments indicate  that  the  compressibility  of  water  diminishes 
as  the  temperature  increases. 

TABLE  OF  ELASTICITY  OF  VOLUME  OF  LIQUIDS. 

(Reduced  from  Grasses  results.) 


Liquid. 

Elasticity  of  Volume. 

Temperature. 

Mercury  .  .. 

717,000,000 

o°      C. 

Water.    ... 

j  42,000,000 
1  45,900,000 

o°      C. 

18°      C. 

Sea-  water.. 

52,900,000 

Ether    . 

j  1  6,  P.  80,000 

o°      C. 

\  15,000,000 

14°      C. 

Alcohol.  .  .. 

(  25.470,000 
(  23,380,000 

7.3°oC. 
13-1    C. 

Oil      

44,090,000 

N.  B. — The  value  for  mercury  is  probably  erroneous. 

If  a  volume  Fof  a  fluid  is  compressed  by  an  amount  AV 
under  an  increase  Ap  of  the  pressure,  then 


" 

4  HYDRA  ULICS. 

AV 

is  called  the  cubical  compression,  and 

V— --  is  termed  the  elasticity  of  volume.     This  is  sensibly 

constant. 

The  vertical  distance  between  the  free  surface  of  a  mass  of 
water  and  any  datum  plane  is  called  the  head  with  respect  to 
that  plane.  If  the  water  extends  down  to  the  level  of  the 
plane,  a  pressure/  is  produced  at  that  level,  and  the  value  of  pr 
so  long  as  the  water  is  at  rest,  is  given  by  the  equation 

^  =  A+4, 

u  -fir.J^ 

w  being  the.  specific  weight  of  the  water  and  /0  the  pressure 
at  the  free  surface.  Thus  the  pressure  may  be  measured  in 
terms  of  the  head,  and  hence  the  expression  "head  due  to 
pressure  or  pressure  head." 

The  mean  value  of  the  atmospheric  pressure  is  14.7  Ibs.  per 
square  inch. 

A  ,       ,    (  is  equivalent  to 

A  head  Of  a  pressure  of 

2.3  ft.  of  fresh  water I  Ib.  per  sq.  in. 

2.25  ft.  of  salt  water I  Ib.  per  sq.  in. 

About  34  ft.  of  fresh  water 14.7  Ibs.  per  sq.  in. 

"       33  ft.  of  salt         "      14.7  Ibs.  per  sq.  in. 

A  head  of  water  is  a  source  of  energy.  A  volume  of  water 
descending  from  an  upper  to  a  lower  level  may  be  employed 
to  drive  a  machine  which  receives  energy  from  the  water  and 
utilizes  it  again  in  overcoming  the  resistances  of  other  machines 
doing  useful  work. 

Let  Q  cu.  ft.  of  water  per  second  fall  through  a  vertical 
distance  of  1i  ft.  Then  the  total  power  of  the  fall  =  wQIi 

ft. -Ibs.  —  —=—    h.   p.,   w   being   the   weight    of  the    water   in 

pounds  per  cubic  foot. 

Let  K  be  the  proportion  of  the  total  power  which  is 
absorbed  in  overcoming  frictional  and  other  resistances.  Then 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  5 

the  effective  power  of  the  fall  =  ze/(2^  (r  ~  ^0>  and  the  efficiency 
is  i  -  AT. 

Imagine  a  bounding  surface  enclosing  a  space  of  invariable 
volume  in  the  midst  of  a  moving  mass  of  fluid.  The  principle 
of  continuity  affirms  that  in  any  interval  of  time  the  flow  into 
the  space  must  be  equal  to  the  outflow  during  the  same  inter- 
val. Giving  the  inflow  a  positive  and  the  outflow  a  negative 
sign,  the  principle  may  be  expressed  symbolically  by 

=  o. 

The  continuity  of  a  mass  of  water  will  be  preserved  so  long 
as  the  pressure  exceeds  the  tension  of  the  air  held  in  solution. 
It  is  on  account  of  the  pressure  of  this  air  that  pumps  cannot 
draw  water  to  the  full  height  of  the  water  barometer,  or  about 
34  ft. 

Generally  speaking,  the  pressure  at  every  point  of  a  contin- 
uous fluid  must  be  positive.  A  negative  pressure  is  equivalent 
to  a  tension  which  will  tend  to  break  up  the  continuity  pre- 
supposed by  the  formulae  ;  and  should  negative  pressures  result 
from  the  calculations,  the  inference  would  be  that  the  latter 
.are  based  upon  insufficient  hypotheses. 

The  pressure  in  water  flowing  through  the  air  cannot  at 
any  point  fall  below  the  atmospheric  pressure.  There  are  cases, 
however,  as  in  water  flowing  through  a  closed  pipe  (Art.  3, 
Chap.  Ill),  in  which  the  pressure  may  fall  below  this  limit  and 
become  almost  nil.  But  there  is  then  a  danger  of  the  air  held 
in  solution  being  set  free,  thus  tending  to  interrupt  the 
•continuity  of  the  flow,  which  may  be  wholly  stopped  if  the  air 
is  present  in  sufficient  volume. 

Consider  a  length  of  a  canal  or  stream  bounded  by  two 
normal  sections  of  areas  Alf  At,  and  let  vlt  v^  be  the  mean 
normal  velocities  of  flow  across  these  sections.  Then  by  the 
principle  of  continuity 


and  the  velocities  are  inversely  as  the  sectional  areas. 

Again,  assume  that  a  moving  mass  of  fluid  consists  of  an 


O  HYDRA  ULICS. 

infinite  number  of  stream-lines,  and  consider  a  portion  of  the 
mass  bounded  by  stream-lines  and  by  two  planes  of  areas  Alt 
AI  at  right  angles  to  the  direction  of  flow.  If  v^  ,  ^2  are  the 
mean  velocities  of  flow  across  the  planes, 

V^AI  =•  Q  =  VyA9  if  the  fluid  is  incompressible. 

Assuming  that  the  fluid  is  compressible,  and  that  the  mean 
specific  weights  at  the  two  planes  are  wl  and  w9  ,  then  the 
weight  of  fluid  flowing  across  Al  is  equal  to  the  weight  which 
flows  across  A^  ,  since  the  weight  of  fluid  between  the  two 
planes  remains  constant.  Hence 


4.  Bernouilli's  Theorem.  —  This  theorem  is  based  on  the 
following  assumptions  : 

(1)  That  the  fluid  mass  under  consideration  is  a  steadily 
moving  stream  made  up  of  an  infinite  number  of  stream-lines 
whose  paths  in  space  are  necessarily  fixed. 

(2)  That  the  velocities  of  consecutive  stream-lines  are  not 
widely  different,  so  that  viscosity,  or  the  frictional  resistance 
between  the  stream-lines,  is  sufficiently  small  to  be  disregarded. 

(3)  That  the  fluid  is  incompressible,  so  that  there  can  be  no 
internal  zvork  due  to  a  change  of  volume. 

In  any  given  stream-line  let  a  portion  AB,  Fig.  I,  of  the 
fluid  move  into  the  position  A'B'  in  /  seconds. 


B  B' 


i' ' 


FIG.  i. 


Let  al ,  pl ,  vl ,  zl  be  the  normal  sectional  area,  the  intensity 
of  the  pressure,  the  velocity  of  flow,  and  the  elevation  above 


FLO  W   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  / 

a  datum  plane  ZZ  of  the  fluid  at  A.     Let  tfa,/3,  z>2,  z^  denote 
similar  quantities  at  B. 

Since  the  internal  work  is  nil,  the  work  done  by  external 
forces  must  be  equivalent  to  the  change  of  kinetic  energy. 

Now  the  external  work 
=  the  work  done  by  gravity  -f-  the  work  done  by  pressure. 

But  when  the  fluid  AB  passes  into  the  position  A'ft  ',  the 
work  done  by  gravity  is  equivalent  to  the  work  done  in  the 
transference  of  the  portion  BB'  ,  and  therefore,  t  beng  the 
time. 


the  work  dw  by  gr^^ty  =  wa^AA'-z^  —  wa^-BB'  ' 


=  wQt  (*>-*,), 

since  A  A'  =  vj,  BB'  =  vj,  and  alvl  =  Q  =  a.^- 

Again,  the  work  done  by  the  pressures  on  the  ends  A  and  B 


The  work  done  by  the  pressure  on  the  surface  of  the  stream- 
line between  A  and  B  is  nil,  since  the  pressure  is  at  every  point 
normal  to  the  direction  of  motion. 
The  change  of  kinetic  energy 

=  kinetic  energy  of  A'  B'  —  kinetic  energy  of  AB 
=  kinetic  energy  of  BB'  —  kinetic  energy  of  AA'  , 
since  the  motion  is  steady,  and  there  is  therefore  no  change  in 
the  kinetic  energy  of  the  intermediate  portion  A'  B.     Thus, 


w  v         w  V 

the  change  of  kinetic  energy  =  -  a^BB'^-  --  a.  A  A— 


w 


Hence,  equating  the  external  worl{  and  the  change  of  kinetic 
energy, 

«>Qt  (*,  -  *,)  +  &  (A  ~  A)  =      &        --        , 


8  HYDRA  ULICS, 

which  may  be  written  in  the  form 

w  v?  ,   w  v?  .  . 

««,+/>,  +  -  -y  =  ^,+A  +  -->    ...    (i) 


But  A  and  B  are  arbitrarily  chosen  points,  and  therefore, 
at  any  point  of  a  stream-line,  the  motion  being  steady  and 
the  viscosity  nil,  the  gradual  interchange  of  the  energies  due 
to  head,  pressure,  and  velocity  is  expressed  by  the  equation 

w  V*  fa  i 

W2  j_  p  _L =  wH,  a  constant ;   /   ...     .     (3) 

~r  r    \    g    2  VJ/ 

+/     I  Is,         **   I  •  V^     \M 

frb*'  f  -  vXm  r-  J  V        -i 

or  n  -iVj  /  / 


z  being  the  elevation  fef  thevpoint  above  the  datum  line,  /  the 
pressure  at  the  point,  w  the  specific  weight,  and  v  the  velocity 
of  flow.  This  is  Bernouilli's  theorem. 

Thus  the  total  constant  energy  of  wH  ft.-lbs.  per  cubic  foot 
of  fluid,  or  H  ft.-lbs.  per  pound  of  fluid,  is  distributed  uniformly 
along  a  stream-line,  wH  being  made  up  of  wz  ft.-lbs.  due  to 

w  z? 
head,/  ft.-lbs.  due  to  pressure,  --  ft.-lbs.  due    to  velocity, 

and  H  being  made  up  of  z  ft.-lbs.  due  to  head,  —   ft.-lbs.  due 

v* 
to  pressure,  and  —  ft.-lbs.  due  to  velocity. 

Assuming  that 

(a)  the  motion  is  steady, 

(ft)  the  frictional  resistance  may  be  disregarded, 

(c)  the  fluid  is  incompressible, 

Bernouilli's  theorem  may  be  applied  to  currents  of  finite  size 
at  any  normal  section,  if  the  stream-lines  across  that  section 
are  sensibly  rectilinear  and  parallel.  There  is  then  no  interior 
work  due  to  a  change  of  volume,  and  the  distribution  of  the 
pressure  in  the  section  under  consideration  will  be  the  same  as 


FLO  W    THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  9 

if  the  fluid  were  at  rest,  that  is,  in  accordance  with  the  hydro- 
static law.  This  is  also  true  whether  the  flow  takes  place  under 
atmospheric  pressure  only,  or  whether  the  fluid  is  wholly  or 
partially  confined  by  solid  boundaries,  as  in  pipes  and  canals, 
or  whether  the  flow  is  through  another  medium  already  occu- 
pied by  a  volume  of  the  fluid  at  rest  or  moving  steadily  in  a 
parallel  direction.  In  the  last  case  there  must  necessarily  be 
a  lateral  connection  between  the  two  fluids,  but  the  pressure 
over  the  section  must  follow  the  hydrostatic  law  throughout 
the  separate  fluids,  and  there  can  be  no  sudden  change  of 
pressure  at  the  surface  of  separation,  as  this  would  lead  to  an 
interruption  of  the  continuity. 

The  hypotheses,  however,  upon  which  these  results  are  based 
are  never  exactly  realized  in  actual  experience,  and  the  results 
can  only  be  regarded  as  tentative.  Further,  they  can  only  ap- 
ply to  an  indefinitely  short  length  of  the  current,  as  the  viscosity, 
-which  is  proportional  to  the  surface  of  contact,  would  other- 

wise become  too  great  to  be  disregarded. 

5.  Applications.  —  If  a  glass  tube,  open  at  both  ends,  and 

called  a  piezometer  (TrieCeiv,  to   press  ;   jterpor,    a  measure) 

is  inserted   vertically   in  the    cur- 

rent,   Fig.    2,   at  a  point  N,  z  ft. 

above  the  point  O  in  the  datum 

line,   the  water  will   rise    in    the 

tube  to  a  height  MN  dependent 

upon    the    pressure   at   N.      The 

effect  of  the  eddy  motion  produced 

at  N  by  obstructing  the  stream- 

lines may  be  diminished  by  mak- 

ing this  end   of  the  tube  parallel 

to  the  direction  of  flow.     Neglect- 

ing  altogether   the   effect  of    the  o 

eddies,  and  taking/  to  be  the  in-  FlG-  2- 

tensity  of  the  pressure  at  TV,  and/0  the  intensity  of  the  atmos- 

pheric pressure,  then, 


w  w 


10 

and  therefore 


HYDRA  ULICS. 


w  w 


=  ON  +  MN  +  - 
1    w 


=  Q  M  +  -. 

1    w 


(5) 


The  locus  of  all  such  points  as  M  is  often  designated  "  the 
line  of  hydraulic  gradient,"  or  the  "  virtual  slope,"  terms  also- 
used  when  friction  is  taken  into  account. 

Let  the  two  piezometers  AB,  CD,  Fig.  3,  be  inserted  in  the 
current  at  any  two  points  B  and  D,  z^  ft.,  and  z%  ft.  respect- 
ively above  the  points  E  and  F  in  the  datum  line. 


FIG.  3. 

Let  /,  be  the  intensity  of  the  pressure  at  B  in  pounds  per 
square  foot,  /2  that  at  D,  and  let  the  water  rise  in  these  tubes 
to  the  heights  BA,  DC.  Then 


w 
and  therefore 


=  zl+,     and       l 

w  w 


w 


•  +        -       +        =^-^=^'    .    .    (6) 
the  line  AG  being  parallel  to  the  datum  line. 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC. 


II 


Thus 


,  (zl  -|-  —  J  —  Ls2  +  —  J  is  equal  to  the  fall  of  the  free 

surface  level  between  the  points  B  and  D. 

Let  vl ,  7'2  be  the  velocities  of  flow  at  B  and  D.     Then  by 
Bernoulli's  theorem 


W          2g  W          2g 

and  therefore  the  fall  of  free  surface  level  between  B  and  D 


(7) 


W 


2g 


Equation  (7)  may  also  be  written  in  the  form 

V?  V?     ,      I         ,     Pi\  (         .      Pl\  V?     .     rr  SQ\ 

-  = r  (Zi  H J  —  l*i  H }  = r  £k»    .    (8) 

2g         2g         ^  W'          *  W'          Zg 

so  that  the  velocity  at  D  is  equal  to  that  acquired  by  a  body 
with  an  initial  velocity  vl  falling  freely  through  the  vertical 
distance  CG. 

Froude  illustrated  Bernouilli's  theorem  experimentally  by 
means  of  a  tube  of  varying  section,  Fig.  4,  conveying'  a  current 


FIG.  4. 

between  two  cisterns.     The  pressure  at  different  points  along 
the  tube  is  measured  by  piezometers,  and  it  is  found  that  the 


12  HYDRAULICS. 

water  stands  higher  and  the  pressure  is  therefore  greater,  where 
the  cross-section  is  larger  and  the  velocity  consequently  less. 
If  the  section  of  the  throat  at  A  is  such  that  the  velocity  is 
that  acquired  by  a  body  falling  freely  through  the  vertical  dis- 
tance h  between  A  and  the  surface  level  of  the  water  in  the 
cistern,  and  if  /  be  the  pressure  at  A,  and  z  the  elevation  of  A 
above  datum,  then,  neglecting  friction, 


W         2g  W 

But  v*  =  2gh,  and  therefore  /  =  pQ  ,  so  that  the  pressure  at 
A  is  that  due  to  atmospheric  pressure  only.  Thus,  a  portion 
of  the  pipe  in  the  neighborhood  of  A  may  be  removed,  as  in 
the  throat  of  the  injector. 

Again,  let  the  cross-section  in  the  throat  at  B  be  less  than 
that  at  A.  The  pressure  at  B  will  be  less  than  the  atmospheric 
pressure,  and  a  column  of  water  will  be  lifted  up  in  the  curved 
piezometer  to  a  height  k'  . 

Let  tf  ,  ,  zl  ,pl  ,  vl  be  the  sectional  area,  elevation  above  datum, 
pressure,  and  velocity  at  B. 

Let  #3  ,  z^  ,pi  ,  z>a  be  similar  symbols  at  E. 

Then 

I  J  ,,  +  A  +  ^+A      ^  =  ,  l  +  A_*'  +  !i2.     (9) 

V  W         2g  W     '    2g  W  '    2g 

Put  //,  =  #,-[-  —  -,  the  height  above  datum  to  which  the 

w 

water  is  observed  to  rise  in  the  piezometer  inserted  at  E,  and 

also  let  #;=*,  +  A  -  h'.     Then 

w 


since  ap^  =  alvl  ,  #a  being  the  sectional  area  at  E.     Therefore 


ft.,    —  a, 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  1 3 

an  equation  giving  the  theoretical  velocity  of  flow  at  the  throat 
B.  Hence  the  theoretical  quantity  of  flow  across  the  section 
at  B  is 


-  a 


(10) 


This  is  the  principle  of  the  Venturi  water-meter  and  also  of 
the  aspirator. 

The  actual  quantity  of  flow  is  found  by  multiplying  equa- 
tion (10)  by  a  coefficient  C  whose  value  is  to  be  determined 
by  experiment. 

If  the  pressure  at  E  is  positive,  then  //,  is  merely  the 
height  to  which  the  water  is  observed  to  rise  in  an  ordinary 
piezometer  inserted  at  E. 

Again,  Froude  also  points  out  that  when  any  number  of 
combinations  of  enlargements  and  contractions  occur  in  a  pipe, 
the  pressures  on  the  converging  and  diverging  portions  of  the 
pipe  will  balance  each  other  if  the  sectional  areas  and  directions 
of  the  ends  are  the  same. 

6.  Orifice  in  a  Thin  Plate. — If  an  opening  is  made  in  the 
wall  or  bottom  of  a  tank  containing  water,  the  fluid  particles 


FIG.  6. 


FIG.  7. 


immediately  move  towards  the  opening,  and  arrive  there  with 
a  velocity  depending  upon  its  depth  below  the  free  surface. 
The  opening  is  termed  an  "  orifice  in  a  thin  plate  "  when  the 
water  springs  clear  from  the  inner  edge,  and  escapes  without 
again  touching  the  sides  of  the  orifice.  This  occurs  when  the 


UNIVERSITY 


14  HYDRA  ULICS* 

bounding  surface  is  changed  to  a  sharp  edge,  as  in  Fig.  5,  and 
also  when  the  ratio  of  the  thickness  of  the  bounding  surface 
to  the  least  transverse  dimension  of  the  orifice  does  not  exceed 
a  certain  amount  which  is  usually  fixed  at  unity,  as  in  Figs. 
6  and  7. 

Owing  to  the  inertia  acquired  by  the  fluid  filaments  there 
will  be  no  sudden  change  in  their  direction  at  the  edge  of  the 
orifice,  and  they  will  continue  to  converge  to  a  point  a  little 
in  front  of  the  orifice,  where  the  jet  is  observed  to  contract  to 
the  smallest  section.  This  portion  of  the  jet  is  called  the  vena 
contracta  or  contracted  vein,  and  the  fluid  filaments  flow  across 
the  minimum  section  in  sensibly  parallel  lines,  so  that  here,  if 
the  motion  is  steady,  Bernouilli's  theorem  is  appli- 
c  cable. 

The  dimensions  of  the  contracted  section  and 
F  its  distance  from  the  orifice  depend  upon  the  form 
and  dimensions  of  the  orifice  and  upon  the  head 
of  water  over  the  orifice. 

Let  Fig.  8  represent  the  portion  of  the  jet  be- 
tween a  circular  orifice  of  diameter  AB  and  the 
contracted  section  of  diameter  CD,  EF  being  the  distance 
between  AB  and  CD.  Then,  taking  the  average  results  of  a 
number  of  observations,  it  is  found  that  AB,  CD  and  EF  are 
in  the  ratios  of  100  to  80  to  50. 

Thus  the  areas  of  the  contracted  section  and  of  the  orifice 
are  in  the  ratio  of  16  to  25,  and,  generally  speaking,  this  is 
assumed  to  be  the  ratio  whatever  may  be  the  form  of  the 
orifice. 

7.  Torricelli's  Theorem. — Let  Fig.  9  represent  a  jet  issu- 
ing from  a  thin-plate  orifice  -in  the  side  of  a  vessel  containing 
water  kept  at  a  constant  level  AB. 

Let  XXbz  the  datum  line,  J/A^the  contracted  section,  and 
consider  any  stream-line  mn,  m  being  in  a  region  where  the 
velocity  is  sensibly  zero,  and  n  in  the  contracted  section.  Then 
by  Bernouilli's  theorem,  the  motion  being  steady, 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  1 5 

/,  /,  being  the  pressures  at  n  and  ;«,  and  zt  2l  their  elevations 
above  datum.     Hence 


(2) 


FIG.  9. 
If  the  flow  is  into  the -atmosphere, 

/  —  the  atmospheric  pressure  =  /0 ,  and 
p,  =  w.Om  +/., 

O  being  the  point  in  which  the  vertical  through  m  intersects 
the  free  surface.     Thus, 


—  =  z .  —  z  +  Om  = 
2g 


(3) 


h  being  the  depth  of  n  below  the  free  surface. 

The  result  given  by  equation  (3)  was  first  deduced  by  Tor- 
ricelli. 

The  depth  below  the  free  surface  is  very  nearly  the  same 
for  all  points  of  the  contracted  vein,  and  the  value  of  v  as 
given  by  (3)  is  taken  to  be  the  theoretical  mean  velocity  of 
flow  across  the  contracted  section. 

Equation  (3)  is  equivalent  to  the  statement  that  when  the 
orifice  is  opened  the  hydrostatic  energy  of  the  water,  viz.,  //  ft.- 

7'2 

Ibs.  per  pound,  is  converted  into  the  kinetic   energy  of    — 


10  i  HYDRA  ULICS. 

ft.-lbs.  per  pound.  Thus,  if  the  jet  is  directed  vertically  upwards, 
it  will  very  nearly  rise  to  the  level  of  the  free  surface,  and  would 
reach  that  level  were  it  not  for  air  resistance,  or  for  viscosity,  or 
for  friction  against  the  sides  of  the  orifice,  or  for  a  combination 
of  these  retarding  causes. 

If  the  jet  issues  in  any  other  direction,  it  describes  a  para- 
bolic arc  of  which  the  directrix  lies  in  the  free  surface. 

Let  OTV,  Fig.  10,  be  such  a  jet,  its  direction  at  the  orifice 


FIG.  10. 

at  O  making  an  angle  a  with  the  vertical.  With  a  properly 
formed  orifice  a  greater  or  less  length  of  the  jet  will  have  the 
appearance  of  a  glass  rod,  and  if  this  portion  were  suddenly 
solidified  and  supported  at  the  ends,  it  would  stand  as  an  arch 
without  any  shearing  stress  in  normal  sections. 

Again,  the  horizontal  component  of  the  velocity  of  flow  at 
any  point  of  the  jet  is  constant  (=  v  sin  or),  so  that,  for  the 
unbroken  portion  of  the  jet,  equidistant  vertical  planes  will 
intercept  equal  amounts  of  water,  and  the  height  of  the  C.  G. 
of  the  jet  above  the  horizontal  line  O  V  will  be  two  thirds  of 
the  height  of  the  jet. 

8.  Efflux  through  an  Orifice  in  the  Bottom  or  in  the  Side 
of  a  Vessel  in  Motion. — If  a  vessel  containing  water  zit.  deep 
ascend  or  descend  vertically  with  an  acceleration  /,  the  press- 
ure/ at  the  bottom  is  given  by  the  equation 

w 
±  -*/  =  p  -  p0  -  wzy 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC. 

being  the  atmospheric  pressure.     Therefore 


/-A 

w 


=•( 


« ±4 


If  now  an  orifice  is  opened  at 
the  bottom,  the  velocity  of  efflux  v 
is  still  taken  as  due  to  the  head  of 
the  pressure  /,  and  therefore  by 
Torricelli's  Theorem 


Let    W,   be   the   weight  of   the 

vessel  and  water,  and  let  the  vessel  be  connected  with  a 
counterpoise  of  weight  W^  by  meansof  a  rope  passing  over  a 
pulley.  Then  by  Newton's  second  law  of  motion,  and  neglect- 
ing pulley  friction, 


g~          W,  W,  W.+  Wt' 

T  being  the  tension  of  the  rope. 

Next  let  a  cylindrical  vessel, 
Fig.  12,  of  radius  r  and  containing 
water,  rotate  with  an  angular  veloc- 
ity oo  about  its  axis.  The  surface 
of  the  water  assumes  the  form  of  a 
paraboloid  of  which  the  latus 

2fT 

rectum  is  —  ^.     If  an  orifice  is  made 
GO 

at  Q  in  the  side  of  the  vessel,  the 
water  will  flow  out  with  a  velocity 
v  due  to  the  head  of  pressure  at 


FIG.  12. 
the  orifice.     This  head  is  PQ,  and 


PQ  =  ON  ±  z  = 


CA) 


z  being  the  vertical  distance  OM  between  the  orifice  and  the 
vertex  of  the  paraboloid.     Hence  by  Torricelli's  theorem 


i8 


HYDRA  ULICS. 


*' 


or 


9.  Application  to  the  Flow  through  a  Frictionless  Pipe 
of  Gradually  Changing  Section  (Fig.  13).— Let  the  pipe  be 
supplied  from  a  mass  of  water  of  which  the  free  surface  is  H  ft. 
above  datum. 

Let  al9  pv  vv  be  the  sectional  area,  pressure,  and  velocity  of 
flow  at  any  point  A,  zl  ft.  above  datum  and  h^ 
ft.  below  the  free  surface. 

Let#a,/3,  z/3  be  similar  symbols  for  a  second  point  B,  ^  ft. 
above  datum  and  h^  ft.  below  the  free  surface. 


FIG.  13. 
Then  by  the  condition  of  continuity 

a&i  =  atvt, 
and  by  Torricelli's  theorem 


2g 


/J 


t 


IV 


FLOW    THROUGH  ORIFICES,   OVER   WEIRS,  ETC. 

and 

2g   ~        *"»  W 

Hence 


so  that  Bernouilli's  theorem,  viz., 

—  +  -  +  2  =  &  +—  =  a  constant, 

2£-   '    w    '  '    w 

holds  true  for  the  assumed  conditions. 

10.  Hydraulic  Resistances  —  (a)  Coefficient  of  Velocity.  — 
In  reality,  the  velocity  v  at  the  vena  contracta  is  a  little  less 
than  V2gh  (Art.  7,  eq.  3)  and  the  ratio  of  v  to  V  '  2gh  is  called 
the  coefficient  of  velocity,  and  may  be  denoted  by  cvt  so  that 


v  =  cv 

Again,  the  equations  for  the  velocity  of  discharge  in  the 
case  of  moving  vessels  now  become 


2g 

and 


A  mean  value  of  cv  for  well-formed  simple  orifices  is  .974. 

An  easy  method  of  determining  the  value  of  cv,  experi- 
mentally, may  be  indicated  by  reference  to  the  jet  represented 
in  Fig.  10,  p.  1 6. 

Measure  the  vertical  and  horizontal  distances  from  the 
orifice  of  any  two  points  A,  B  in  the  jet. 

Let  jj>,,  x^  denote  the  co-ordinates  of  A. 

Let  yv  x^  denote  the  co-ordinates  of  B. 

Then  if  tl  is  the  time  occupied  by  a  fluid  particle  in  moving 
from  the  orifice  to  A,  and  t^  the  time  from  the  orifice  to  B, 


2O  HYDRA  ULICS. 


—  v  sin  a  .  tl  ;     j^  =  v  cos  <*  .  /,  --  £'/12  ; 


^  =  v  sin  a  .  /3  ;     J2  =  ^  cos  ar  .  /3  --  gt*. 


x 


,=^  cot  a-        ,    .a     , 
2  z;2  sin2  a 


P- 

=  x   cot  a  —  - 
2 


2  v*  sin3  a' 
By  means  of  the  two  last  equations 


and 


2   sin  a  (xl  cot  a  —  yj 
so  that 

*t 


' 


Hence  />  , 

and 


4^  sin2  a  (xl  cot  a  —  y^  ' 

and  since  the  values  of  x^  yv  x»  y^  are  known,  equation   I 
will  give  the  value  of  a,  and  equation  2  the  value  of  cv. 

Note. — If  the  jet  issues  from  the  orifice  horizontally,  a  =  90°,  and  the 
last  equation  becomes 


so  that  the  position  of  one  point  only  relatively  to  the  orifice  need  be 
observed. 


(b)  Coefficient  of  Resistance. — Let  hv  be  the  head  required 
to  produce  the  velocity  v.  Let  hr  be  the  head  required  to 
overcome  the  frictional  resistance.  Then 

h,  the  total  head,  —  hv-\-  hr  =  hv(i  -}-  cr), 
where  hr  =  trhv. 


FLOW   THROUGH  ORIFICES,   OVER    WEIRS,  ETC.  21 

cr  is  termed  the  coefficient  of  resistance,  and  is  approxi- 
mately constant  for  varying  heads  with  simple  sharp-edged 
orifices.  Again, 


Hence 

and  therefore 


so  that  cr  can  be  found  when  cv  is  known,  and  vice  versa. 

(c)  Coefficient  of  Contraction.  —  The  ratio  of  the  area  a  of 
the  vena  contracta  to  the  area  A  of  the  orifice  is  called  the  co- 
efficient of  contraction,  and  may  be  denoted  by  cc. 

The  value  of  cc  must  be  determined  in  each  case,  but  in 
sharp-edged  orifices  an  average  value  of  cc  ,  as  already  pointed 

out,  is  —  -   =  .64.    C<zteris  paribus,  cc  increases  as  the  orifice  area 

and  the  head  diminish. 

The  following  are  some  of  the  conditions  which  tend  to 
modify  the  value  of  cc  : 

(1)  The   contraction    is  imperfect  and  will    be   suppressed 
over  the  lower  edge  of  a  square  orifice  at  the  bottom  of  a  ves- 
sel, and  over  a  side  as  well  if  the  orifice  is  in  a  corner.     In  fact, 
the  contraction  is  more  or  less  imperfect  for  any  orifice  within 
three  diameters  from  the  side  or  bottom  of  the  vessel.     Thus, 
the   cross-section  of  the  vena  contracta  is   in- 

creased, and   experiment    shows  that   the  dis- 
charge is  also  increased. 

(2)  ce  is  increased  or  diminished  according 
as  the  surface  surrounding  the  orifice  is  convex 
or  concave  to  the  interior  of  the  vessel. 

(3)  The  contraction   is  imperfect  and    ce  is         FIG.  14. 
increased   if  the   orifice  is  placed  in  a  confined 

part  of  the  vessel  or  if  it  approaches  the  orifice  through  a  chan- 
nel, as  in  Fig.  14,  the  velocity  of  the  fluid  filaments  being 
thereby  considerably  increased. 


22 


HYDRA  ULICS. 


(4)  If  the  inner  edge  of  an  orifice  is  rounded,  as  shown  by 
Figs.    15  and   16,  the  contraction  is  more  or  less  imperfect. 


FIG.  15. 


FIG.  16. 


The  value  of  ce  varies  from  .64  for  a  sharp-edged  orifice  to  very 
nearly  unity  for  a  perfectly  rounded  orifice. 

(5)  The  contraction  is  incomplete  when  a  border  or  rim  is 
placed  round  a  part  of  the  edge  of  the  orifice,  projecting  in- 
wards or  outwards.  According  to  Bidone, 


and 


•cc  =  .62(1  +  .152  -)  for  rectangular  orifices, 
ce  =  .62! i  -j-  .128  -)  for  circular  orifices, 


n  being  the  length  of  the   edge  of  the  orifice  over  which  the 
border  extends,  and  p  the  perimeter  of  the    orifice. 

(6)  If  the  sides  of  the  orifice  are  curved  so  as  to  form  a 
bell-mouth  of  the  proportions  shown  by  Fig.    17,  and  corre- 


j* 1-.6 

FIG.  17. 


^ 


spending  approximately  to  the  shape  of  the  vena  contracta, 
the  whole  of  the  contraction  will  take  place  within  the  bell- 


FLO W   THROUGH  ORIFICES,   OVER    WEIRS,  ETC.  2$ 

mouth,  and  cc  is  unity  if  the  area  of  the  orifice  is  taken  to  be, 
the  area  of  the  smaller  end. 

For  such  an  orifice  Weisbach  gives  the  following  table  of 
values  of  c, : 


Head  over  Orifice  in  Feet. 

.66 

i  .64 , 

11.48 

5577 


FIG.  18. 


cv. 

959 

967 

975 

994 

337-93 994 

The  dimensions  of  the  jet  at  the  contracted  section  or  at 
any  other  point  may  be  directly  measured  by 
means  of  set-screws  of  fine  pitch,  arranged 
as  in  Fig.  18.  The  screws  are  adjusted  so  as 
to  touch  the  surface  of  the  jet,  and  the  dis- 
tance between  the  screw-points  is  then  meas- 
ured. 

(d)  Coefficient  of  Discharge. — If  Q  is  the 
quantity  of  flow  per  second  across  the  con- 
tracted section,  then 


—  cccvA  V2gh  =  cA 


where  c  —  cccv  is  the  coefficient  of/discharge,j  and  is  to  be  de- 
termined by  experiment. 

The  values  of  c  in  thousandths  for  orifices  of  different 
forms,  given  in  tables  A  and  B,  have  been  deduced  by  the 
author  from  an  extended  series  of  experiments  carried  out  in 
the  hydraulic  laboratory,  McGill  University. 

The  experimental  tank  is  about  30  ft.  in  height  and  its 
horizontal  section  is  square,  with  an  interior  area  of  25  sq. 
ft.  The  inside  faces  of  the  tank  are  plumb,  and  there  are 
no  projections  to  interfere  with  the  stream-lines. 

The  letters  T  and  S  at  the  head  of  the  columns  respec- 
tively indicate  that  the  orifice  is  in  a  plate  of  thickness  .16  in., 
or  is  sharp-edged. 


HYDRA  ULICS. 


TABLE   A. 
OF  VALUES  OF  c  FOR  ORIFICES  OF  .197  SQ.  IN.  IN  AREA. 


•£  • 

S 

|| 

._ 

Jg 

|K 

15 
'*j  D 

1  5 

•£» 

_u 

a 

o 

OJ  ^ 

<D  c 

u  c  iC 

|| 

u 

ho 

>eo 

JS  °   ' 

>^:s 

>Cc  „ 

H3 

'Cc75 

> 

Q 

s^ 

•S  "^ 

."§-^ 

•"=-•5 

^-  *c5 

rfj 

J3 

^   S  rC 

^  5  •« 

^  3  a 

•^  5  *+-( 

6 

"o 

9 

l.L 

1'; 

|| 

^"fe 

III 

i»  crd 

i^s 

Hi 

s 

3 

1*1 

rt-o 

ca  u 

5-0  4) 

!.•§•- 

•S'O  § 

£ 

£ 
C 

w 

C/3 

F 

4;C75^ 

IK<S 

cS 

•|Kh 

Head 

T 

S 

T 

S 

T 

S 

S 

T 

S 

s 

S 

S 

in  Feet. 

I 

624 

618 

627 

627 

623 

628 

623 

635 

640 

641 

658 

659 

2 

616 

6n 

620 

621 

6I3 

621 

619 

626 

633 

632 

646 

646 

4 

610 

607 

615 

615 

606 

617 

614 

619 

629 

629 

637 

637 

6 

607 

605 

6I3 

613 

604 

614 

612 

616 

625 

627 

634 

633 

8 

606 

604 

612 

612 

603 

612 

612 

614 

625 

625 

631 

63I 

10 

606 

604 

611 

611 

602 

610 

611 

612 

624 

623 

630 

629 

12 

605 

603 

611 

611 

60  1 

610 

611 

611 

622 

622 

627 

626 

14 

604 

603 

610 

610 

600 

610 

609 

6n 

622 

621 

624 

625 

16 

606 

602 

610 

610 

600 

610 

609 

610 

620 

621 

624 

624 

18 

605 

602 

610 

610 

600 

610 

609 

609 

620 

620 

623 

623 

20 

604 

60  1 

609 

609 

600 

610 

609 

602 

620 

620 

622 

622 

TABLE    B. 

OF  VALUES  OF  c  FOR  FOUR  ORIFICES  OF  .0625  SQ.  IN.  IN  AREA,  AND  FOR 
ONE  TRIANGULAR  ORIFICE  OF  .05  SQ.  IN.  IN  AREA. 


Form 
of     - 
Orifice. 

Circular. 

Equilateral 
Triangle  with 
Horizontal 
Base 

Square  with 
Vertical  Sides 

Rectanc 
Vertica 
equal  u 
the  W 

de  with 
Sides. 
>  Twice 
r;^»H 

Rectangle  with 
Vertical  Sides 
equal  to 
Four  Times 

uppermost. 

the  Width. 

Head 
in  Feet. 

T 

S 

T 

S 

T 

S 

T 

S 

T 

S 

I 

678 

620 

657 

631 

643 

627 

662 

640 

688 

67I 

2 

618 

613 

646 

623 

63I 

621 

643 

629 

655 

657 

4 

610 

605 

628 

616 

620 

615 

63I 

620 

642 

643 

6 

607  • 

601 

628 

613 

615 

612 

627 

616 

634 

636 

8 

606 

60  1 

621 

610 

612 

609 

624 

613 

631 

632 

10 

fo4 

600 

6l8 

608 

6I3 

608 

621 

613 

629 

629 

12 

663 

598 

'6l7 

607 

6n 

606 

621 

611 

626 

627 

14 

602 

598 

6l7 

607 

610 

606 

620 

610 

623 

625 

16 

602 

598 

616 

606 

609 

606 

619 

609 

622 

625 

18 

60  1 

597 

6i5 

605 

607 

605 

618 

608 

622 

623 

20 

60  1 

597 

615 

605 

607 

604 

618 

608 

621 

622 

T 


FLO W   THROUGH  ORIFICES,   OVER    WEIRS,  ETC.  2$ 

The  jet  springs  clear  from  the  orifice  in  all  cases  repre- 
sented in  Tables  A  and  B. 

The  following  inferences  may  be  drawn  from  an  inspection 
of  Tables  A  and  B  : 

(1)  The  coefficient  of  discharge  diminishes  as  the  head  in- 
creases, but  at  a  diminishing  rate. 

(2)  The   coefficients   for  the  thick-plate   orifices  are  in   all 
cases  greater  than   the   corresponding  coefficients   for  sharp- 
edged  orifices,  excepting  in  the  case  of  the  longest  rectangular 
orifice  in  Table  B.     Under  a  head    of  I  ft.  the  coefficient  of 
discharge  for  this  orifice  still  exceeds  that  of  the  same  orifice 
with  sharp  edge,  but  for  heads  exceeding    I   ft.  the  coefficient 
seems  to  be  a  little  less,  but  is  practically  the  same.     It  may 
be   noted   that   the  thickness   of  the   plate  is   2.56  times  the 
width  of  the   orifice,  and  the  contraction  for  the  thick-plate 
orifice  is  consequently  increased. 

(3)  The    coefficient   for   rectangular    orifices    seems  to   be 
practically  the  same  whether  the    longest  side  is  vertical  or 
horizontal. 

(4)  The  coefficient  increases  with  the  area  of  the  orifice, 
excepting  when  the  head  is  very  small.     The  coefficient  for 
orifices    of   small   area   then    rapidly   increases,   as   shown    in 
Table  B. 

(5)  With   rectangular   orifices   the  coefficient   increases  as 
the  width  of  the  orifice  diminishes,  i.e.,  as  the  orifice  becomes 
more  elongated. 

The  two  last  results  are  in  accordance  with  similar  results 
deduced  by  Weisbach,  Buff,  and  others. 

The  coefficient  of  discharge  is  modified  when  the  edges  of 
the  orifices  are  not  sharp,  but  have  a  sensible  thickness,  and 
the  formula  giving  the  discharge  may  be  written 

Q  =  cA  J^H> 

H  being  the  depth  of  the  axis  of  the  orifice  below  the  free 
surface. 

II.  Miner's  Inch. — The  miner's  inch  is  a  term  applied  to 
the  flow  of  water  through  a  standard  vertical  aperture,  one 
square  inch  in  section,  under  an  average  head  of  6£  inches. 


26 


HYDRA  ULICS. 


Taking  c  =  .62, 
the  flow  =  Q 


=  .62  A 

=  -62  x 


=  i£  cu.  ft.  per  minute,  approximately. 

The  term  is  more  or  less  indefinite,  as  the  different  companies 
in  disposing  of  water  to  their  customers  do  not  always  use  the 


FIG.  19. 

same  head,  and  the  flow  is  thus  found  to  vary  from  1.36  to  1.73 
cu.  ft.  for  each  square  inch  of  aperture. 

The  aperture  is  usually  2  in.  deep  and  may  be  of  any  re- 
quired width,  Fig.  19.  The  upper  and  lower  edges  of  the 
aperture  are  formed  by  ij-in.  planks,  the  lower  edge  being  2 
in.  above  the  bottom  of  the  channel,  and  the  plank  forming 
the  upper  edge  being  5  to  5^  in.  deep,  so  that  the  head  over 
the  centre  of  the  aperture  is  from  6  to  6^  inches. 
12.  Energy  and  Momentum  of  the  Jet. 

The  energy  of  the  jet  =  wav  —  ft.-lbs.  per  second 


wav 


ft.-lbs.  per  second 


s.  «  ^ 


*•< 


FLOW   THROUGH  ORIFICES,  OVER   WEIRS,  ETC.  2? 

=  wavhc*  ft.-lbs.  per  second 

wavhc*  ,  N 

=  -  -  h.  p.  (horse-power) 


p  (=  wh}  being  the  hydrostatic  pressure  due  to  the  head  h. 


w 


The  momentum  of  the  jet  ~  -  av  .  v  —  wa  -  =  2wakc* 

<£•  o 


and  this  is  equal  to  the  pressure  in  pounds  produced  by  the  jet 
against  a  fixed  plane  perpendicular  to  its  direction.  Neglect- 
ing cv*,  the  thrust  is  double  the  hydrostatic  pressure  due  to 
the  head  h. 

13.  Inversion  of  the  Jet.  —  The  phenomenon  of  the  inver- 
sion of  the  jet  was  first  noticed  by  Bidone,  and  has  been  subse- 
quently investigated  by  Poncelet,  Lesbros,  Magnus,  Lord 
Rayleigh,  the  author,  and  others.  When  a  jet  issues  from  an 
orifice  in  a  vertical  surface,  the  sections  of  the  jet  at  points 
along  its  path  assume  singular  forms  dependent  upon  the 
nature  of  the  orifice. 

Figs.  20  to  27  are  from  photographs  (taken  from  the  same 
point)  of  jets  issuing  under  the  same  head,  viz.,  12  ft.,  from 
orifices  of  different  forms  and  sizes.  The  dimensions  of  these 
jets  are  comparable  with  the  jets  shown  by  Figs.  20  and  21, 
which  are  issuing  from  circular  orifices  of  I  in.  and  J  in. 
diameter,  respectively. 

With  a  square  orifice,  Fig.  22  (side  =  I  in.),  Fig.  23  (side  = 
.443  in.),  and  Fig.  24  (side  =  .25  in.),  the  section  is  a  star  of 
four  sheets  at  right  angles  to  the  sides. 

With  a  triangular  orifice,  Fig.  25  (side  =  .676  in.),  the  sec- 
tion is  a  star  of  three  sheets  at  right  angles  to  the  sides. 

In  general,  with  a  polygonal  orifice  of  n  sides  the  section 
will  be  a  star  of  n  sheets  at  right  angles  to  the  sides. 

Fig.  26  is  a  jet  from  a  rectangular  orifice  (J  in.  X  J  in.),  its 
section  near  the  orifice  being  a  star  of  four  sheets. 

Fig.  27  is  a  jet  from  a  semi-circular  orifice  (diar.  —  .388  in.), 


FIG.  20. 


FIG.  21. 


FIG.  22. 


FIG.  23. 


FIG.  24. 


FIG.  25. 


Fro.  26. 


FIG.  27. 


FLOW    THROUGH   ORIFICES,   OVER    WEIRS,  ETC.  2Q 

the  section  near  the  orifice  being  a  rounded  boundary  and  a 
single  sheet  at  right  angles  to  the  diameter. 

The  changes  in  the  form  of  the  jet  are  doubtless  due  to  the 
mutual  action  between  the  fluid  particles.  A  filament  issuing 
horizontally  and  freely  at  B,  Fig.  28,  has  a  velocity  2g .  AB,  and 


FIG.  28. 

describes  a  certain  parabola  BD.  A  filament  issuing  horizon- 
tally and  freely  at  a  lower  level  C  has  a  velocity  2g .  AC,  and 
describes  a  parabola  CD  of  less  curvature  than  BD.  Now  the 
two  filaments  cannot  pass  simultaneously  through  the  point  of 
intersection  D,  and  must  necessarily  press  upon  each  other- 
They  are  thus  deviated  out  of  their  natural  paths,  and  the  jet 
spreads  out  into  sheets,  as  described  above. 

If  the  orifice  is  small  and  the  head  not  large,  the  jet,  on 
leaving  the  contracted  section  at  the  orifice,  spreads  out 
into  sheets,  and  then  diminishes  to  a  contracted  section  similar 
to  the  first,  after  which  it  again  spreads  out  into  sheets,  bisect- 
ing the  angles  between  the  first  set  of  sheets,  and  again  dimin- 
ishes to  a  contracted  section.  This  action  is  repeated  so  long 
as  the  jet  remains  unbroken. 

14.  Emptying  and  Filling  a  Canal  Lock.— When  the 
head  varies,  as  in  filling  or  emptying  a  reservoir  or  a  lock,  in 
filling  a  vessel  by  means  of  an  orifice  underwater,  or  in  empty- 
ing water  out  of  a  vessel  through  a  spout,  Torricelli's  theorem 
is  still  employed. 

If  the  lock  or  vessel  is  to  be  filled,  Fig,  29,  let  X  sq.  ft.  be 
the  area  of  the  water-surface  when  it  is  x  ft.  below  the  surface 
of  the  outside  water. 


3°  HYDRA  ULICS. 

If  the  lock  or  vessel  is  to  be  emptied,  Fig.  30,  then  X  sq.  ft. 
is  the  area  of  the  water-surface  when  it  is  x  ft.  above  the  orifice. 


FIG.  29. 


FIG    30. 


In  each  case  JIT  ft.  is  the  effective  head  over  the  orifice,  and 
is  the  head  under  which  the  flow  takes  place. 

In  the  time  dt  the  water-surface  in  the  lock  or  vessel  will 
rise  or  fall  by  an  amount  dx.  Then 

—  A  .dx  =  quantity  which  has  entered  the  lock 
=  cA  <J~2gx  .  dt, 


A  being  the  area  of  the  orifice. 
Hence 


t  = 


Xdx 


cA 


an  equation  giving  the  time  of  filling  or  emptying  the  lock 
between  the  level  x  and  h.  The  value  of  c  for  submerged 
orifices  seems  to  be  somewhat  less  than  when  the  flow  occurs 
freely,  but  it  is  usual  to  take  .6  or  .625  as  a  mean  value. 

15.  General  Equations. — Bernoulli's  theorem  may  be 
easily  deduced  from  the  general  equations  of  fluid  motion,  as 
follows: 

Let/  be  the  pressure  and  p  the  density  at  any  point  whose 
co-ordinates  parallel  to  the  axes  are  x,  y,  2. 

Let  «,  v,  w  be  the  velocities  of  flow  at  the  same  point 
parallel  to  the  axes,  and  let  X,  Y,  Z  be  the  accelerating  forces. 
Then  three  equations  result  from  the  principle  of  the  equality 
of  pressure  in  all  directions,  viz. : 


FLOW   THROUGH  ORIFICES,   OVER    WEIRS,  ETC.  3! 

I  dp  d(u)  du          du          du  du 

-p-dX^x--^r  =  x—dt-uTx-v^-w^  <o 

I  dp  d(v)  dv          dv  dv  dv 

pdj=Y     ^i=Y-7t-uTx-v-dj-w-dz'   & 

I  dp  d(w)  dw          dw         dw  dw 

~j~  =  Z j'  =  Z T~  ~  u  3 v  ~i w  ~r\  fa} 

pdz  dt  dt  dx          dy  dz'   u; 

If  the  motion  is  steady,  so  that  the  velocity  at  any  point  is 

r         ^  r  4.1,  v  1       4.U        du  dv          dw 

a  function  of  the  position  only,  then  ^-—  =  o  =  -=-  —  —  and 

dt  dt         dt 

u,  v,  w  may  be  expressed  as  the  differential  coefficients  of  a 
function  F.     Thus, 

dF  dF  dF 

u  =  —j-',     v  =  —j—',     w  =  -7-; 
dx  dy  dz 

and  therefore 

du         d*_F_    _  ^ 
dy  ~~  dydx  ~~  dx' 

du        d*F    _  dw 

dz    .    dzdx  ~~  dx  ' 

dv  _    d*F    _  dw 
dz  ~~  dzdy  ~~  dy ' 

Hence  equations  i,  2,  and  3  may  be  written 

I  dp  du          dv  dw 

~~:r—X—u-j v-j w  -7- ;     ...    (4) 

p  dx  dx          dx  dx 

i  dp  du  dv  dw 

--^-=Y—u-j v-r  —  w -7-;      .     .     .     (5) 

p  dy  dy  dy  dy' 

i  dp  du  dv       '     dw 

r-  =  Z  —  u  —, v  —r  —  w  —jr.     .     .     .     (6) 

p  dz  dz  dz  dz  ^  } 


32  HYDRAULICS. 

Multiplying  eq.  4  by  dx,  eq.  5  by  dy,  and  eq.  6  by  dz,  and 
adding,  then 


r  dy  '    '    dz 

Idv  ,      .   dv  dv     \ 

—  vi-j-dx  4-  -j-dy  +  -rdz\ 

\dx        '    dy  J    '    dz     / 

/        ,   dw  dw 


which  may  be  written 

dp 
-—  Xdx-\-  Ydy  +  Zdz  —  (udu  +  v  dv  +  wdw). 

Integrating,  and  assuming  the  fluid  to  be  homogeneous, 

u*  4-  v*  4-  wz 
-         —^-  —  +  a  constant. 

Hence,  if  gravity  is  the  only  force,  and  if  V  is  the  resultant 
velocity  at  the  point,  then 


and  the  last  equation  becomes 

P  C  V* 

-  —  —  J  gdz  ---  \-  a  constant 


and  therefore 


=  —  gz  —  —  +  a  constant ; 


p         Fa 

z  -\ -I =  a  constant. 

Pg       ^g 


16.  Loss  of  Energy  in  Shock. — An  abrupt  change  of  sec- 
tion at  any  point  in  a  length  of  piping  destroys  the  parallelism 
of  the  fluid  filaments,  breaks  up  the  fluid,  and  energy  is  dis- 
sipated in  the  production  of  eddy  and  other  motions.  The 
energy  thus  wasted  is  termed  "  energy  lost  in  shock." 


FLOW   THROUGH  ORIFICES,   OV. 


33 


In  a  short  length  of  piping,  where  the  section  suddenly 
changes  from  A'B'  to  EF,  consider  the  fluid  mass  between  the 
two  transverse  sections  AB,  where  the  motion  of  the  fluid  fila- 


FIG.  31. 

ments  has  been  undisturbed  and  is  in  parallel  lines,  and  CD* 
where  the  parallelism  has  been  again  re-established. 

In  an  indefinitely  short  interval  of  time  t  let  the  mass  move' 
forward  into  the  position  bounded  by  the  plane  sections  A '  R" 
and  CD'. 

Let  #„  2/,,  /\  be  the  sectional  area,  velocity  of  flow,  and  mean 
intensity  of  pressure  at  A'B'. 

Let  a9,  ^,,  A   be  similar  symbols  for  CD'. 

Let  z, ,  <8-2  be  the  elevation  above  datum  of  the  C.  G.s  of 
the  sectional  areas  at  A'B'  and  CD'. 

Let //be  the  vertical  distance  between  the  C.  G.s  of  the  areas 


Let  P  be  the  mean  intensity  of  pressure  over  the  annular 

surface  between  £F  and  A'B'. 

The  resultant  force  acting  in  the  direction  of  motion  upon 
the  mass  of  fluid  under  consideration 

=  component  of  weight  of  mass  in  this  direction 
-f-  pressure  on  A'B' 

-[-pressure  on  annular  surface  between  EF  and  A'B' 
—  pressure  on  CD' 


34  HYDRAULICS. 

~  l 


assuming  that  P  =  fil,  or  that  the  mean  intensity  of  pressure  is 
unchanged  throughout  the  whole  of  the  section  EF. 

The  normal  reaction  of  the  pipe-surface  between  EF  and 
CD'  has  no  component  in  the  direction  of  motion,  and  fric- 
tional  resistances  are  disregarded. 

Hence  the  impulse  of  the  resultant  force 


(p,  -  A)  / 

=  change  of  momentum  in  the  same  direction 
of  the  fluid  masses  CDD'C'  and  ABB'  A', 
since  the  momentum  of  the  mass  between 
A'B'  and  CD  remains  unchanged 

w  w 

=  -a,v,.v,t--alv,.vlt 

IV 

=  -  a&S  -  v^t, 

o 

since  by  the  condition  of  continuity 

alvl  =  a^. 

Dividing  throughout  by  the  factor  waj,  the   equation  be- 
comes 

*    ,/,      A      <       ^. 

z.  —  z«  —  n  H  —  —  —  =  —  —  --  , 
1  w      w       g        g 

which  may  be  written  in  the  form 


Now  the  pipes  are  nearly  always  axial,  and  in  such  case 
h  =  o,  so  that  the  last  equation  becomes 

.A  ,  V=       -  A  i  ».'  ,  (».-».)' 

'~*~W~r2g     '    *~f~W~r2g~T          2g 


FLO W   THROUGH  ORIFICES,  OVER   WEIRS,  ETC. 


35 


If  there  had  been  no  abrupt  change  of  section,  or  if  the 
change  between  A  '  B'  and  CD  had  been  gradual,  then  no  in- 
ternal work  would  have  been  done  in  destroying  the  parallelism 
of  the  fluid  filaments,  and  no  energy  wasted.  Therefore,  by 
Bernoulli's  theorem,  the  relation 


would  have  held  good. 

Thus,  ^-2  --  -  ft.-lbs.   of  energy  per  pound  of  fluid  is  the 

loss  in  shock  between  A'  B'  and  CD. 

Experiment  justifies  the  assumption  P  =  pl. 

17.  Mouthpieces.  —  (a)  Bor  das  Mouthpiece.  —  This  is  merely 
a  short  pipe  projecting  inwards,  as  in   Fig.  32,  representing 


FIG.  32. 


a  jet  flowing  through  a  re-entrant  mouthpiece  of  sectional 
area  A,  fixed  in  the  vertical  side  of  a  vessel  of  constant  hori- 
zontal section  and  containing  water  kept  at  a  constant  level. 
The  mouthpiece  is  as  long  as  will  allow  of  the  jet  springing 
clear  from  the  end  EF  without  adhering  to  the  inside  surface. 
The  velocity  of  the  fluid  molecules  along  AC  and  DK  is 
sufficiently  small  to  be  disregarded,  so  that  the  pressure  over 
this  portion  of  the  vessel  is  distributed  in  accordance  with 


36  HYDRAULICS. 

the  hydrostatic  law.     The  same  may  also  be  said  of  the  pressure 
on  the  remainder  of  the  vessel's  surface. 

Again,  the  only  unbalanced  pressure  is  that  on  the  surface 
HG  immediately  opposite  the  mouthpiece,  and  the  resultant 
horizontal  force  in  the  direction  of  the  axis  of  the  mouthpiece 

=  (A  +  wti)A  —  pQA  =  whA, 

h  being  the  depth  of  the  axis  below  the  water-surface  and  /a 
the  intensity  of  the  atmospheric  pressure. 

The  jet  converges  to  a  minimum  or  contracted  section  MN 
of  area  a. 

In  a  unit  of  time  let  the  fluid  mass  between  AB  and  MN 
take  up  the  position  bounded  by  A  'B  and  M' N' .     Then 
whA  =  impulse  of  force  in  direction  of  motion 

=  change  of  momentum  in  same  direction  in  a  unit 

of  time. 

=  difference   between   the   momenta    of    MNN' Mr 
and  ABB'A',  since  the  momentum  of  the  mass 
between  A' B'  and  MN  remains  unchanged 
=  momentum  of  MNN' M' ,  since  the  momentum  of 
ABB' A'  is  vertical 

w  w 

=  —  av .  v  =  —  av  , 
g  g 

v  being  the  mean  velocity  of  flow  across  the  contracted  section. 
Hence 

w  w 

whA  =  —  av  =  —  a.  2gh, 

g  g 

and  therefore 

A  —  2a, 
or 

1  a 

-  =  -j  =  coefficient  of  contraction. 

2  A 

This  result  has  been  very  closely  verified  by  experiment,  the 
coefficient  having  been  found  to  be  .5149  by  Borda,  .5547  by 
Bidone,  and  .5324  by  Weisbach. 


FLOW  THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  37 

Borda's  mouthpiece  gives  a  smaller  discharge  than  a  sharp- 
edged  orifice,  but  a  discharge  which  is  much  more  uniform,  and 
hence  it  is  generally  used  in  vessels  from  which  water  is  to  be 
distributed  by  measure. 

Note. — Let  Fig.  33  represent  a  jet  flowing  through  a  re- 
entrant mouthpiece  of  sectional  area  A  fixed  in  the  sloping 
side  of  a  reservoir  containing  water  kept  at  a  constant  level,  and 
suppose  that  the  reservoir  is  of  such  size  that  GHKL  may 
represent  a  cylindrical  fluid  mass  coaxial  with  the  mouthpiece 
and  so  large  that  the  velocity  at  its  surface  is  sensibly  nil. 
Let  ti,  h  be  the  depths  below  the  water-surface  of  the  C.  G.'s 
of  the  areas  GH  and  KL,  respectively. 


T7\.i 


FIG.  33- 

Then  the  resultant  force  along  the  axis  of  the  mouthpiece 

—  pressure  on  GH  —  pressure  on  CK  and  on  DL 

—  pressure  on  EF 

-f-  component   of   the  weight  of  the  fluid 
mass  GHKL 

—  (po  _[_  whf)  area  GH  —  (p0  +  wJi)  (area  CK '  +  area  DL) 
—  p,  .  area  EF  +  w  .  area  GH  .  GK .  -75^-1  very  nearly 

LrK. 

—  whA. 


38  HYDRA  ULICS. 

Hence,  in  a  unit  of  time, 

whA  =  impulse  of  this  force 

=  change  of  momentum  in  direction  of  axis 

w  w  w 

—  —av  .  v  =  —av  =  —  a  .  2gh, 

g  g  g 

a  being  the  area  of  the  contracted  section,  while  h  is  also  very 
approximately  the  depth  of  its  C.G.  below  the  water-surface. 
Thus,  as  before, 

the  coefficient  of  contraction  =  —  =  -. 

A       2 

(b)  Ring-Nozzle. — The   ring-nozzle   (see    Fig.    34)  is   often 

used  with  a  fire-engine  jet,  and 
consists  of  a  re-entrant  pipe  of 
sectional  area  #,  fixed  in  a  pipe 
of  sectional  area  av  The  length 
of  the  re-entrant  portion  is  such 
that  the  water  springs  clear  from 
the  inner  end  and,  without  again 
touching  the  surface  of  the 
.  mouthpiece,  converges  to  a  mini- 


FIG.  34  mum   or  contracted  section  of 

area  a  at  MN. 

Consider  the  fluid  mass  between  MN  and  a  transverse  sec- 
tion AB,  and  in  a  unit  of  time  let  it  move  into  the  position 
bounded  by  the  planes  MN1  and  A'B'. 

It  is  assumed  that  the  motion  is  steady  and  that  there  is  no- 

internal  work  due  to  the  production  of  eddies  or  other  motions. 

Let  />0 ,  v  be  the  intensity  of  the  atmospheric  pressure  and 

the  velocity  at  MN. 

Let  p^ ,  vl  be  the  mean  intensity  of  pressure  and  the  veloc- 
ity at  AB. 

Let  P  be  the  mean  intensity  of  the  pressure  over  the  annu- 
lar surface  EF,  GH. 

Let  #0 ,  2l  be  the  elevations  above  datum  of  the  C.  G.s  of 
the  sections  MN  and  AB. 


FLO  W   THROUGH  ORIFICES,  OVER   WEIRS,  ETC.  39 

Then 

ow.fo  -  *.)  +  P&  -  P(a,  -  a,)  —p.  a, 


impulse  in  direction  of  motion 

change  of  momentum  in  same  direction  in  a  unit  of  time 
difference  of  the  momenta  of  the  fluid  masses  MNN'M'  and 
ABB'  A' 

~(av*  -  a,*,")- 

Assuming  that  P  =  piy  the  last  equation  becomes 

IV 

wa,(2l  -  *.)  +  a,(p,  -  p.)  =  —(av*  —  a,v*).    .     .    (i) 

o 

By  Bernoulli's  theorem, 

A       *>?  p.       v* 


and  therefore 


W  2g 


Now  s,  —  z0  is  very  small  and  may  be  disregarded  without 
sensible  error,  and  then  by  eqs.  (i)  and  (2) 


v*  —  v?  _  pl  —  p0  _  i  av*  — 
2g  w       ~  g         a, 

Hence 
2 


_ 
a   ~  av*  —  av    ~~  aa    —  a*av*  ~~  a 


since  a9vt  =  av. 

If  the  sectional  area  #a  of  the  pipe  is  very  large  as  compared 

with  a,  so  that  --  may  be  disregarded  without  sensible  error, 

then    —  =  -,   and  therefore    the   coefficient    of    contraction 
a,      a 

=  —  =  -,  as  before. 


HYDRA  ULICS. 


(c]  Cylindrical    Mouthpiece. — Whe-n    water    issues   from    a 
cylindrical  mouthpiece  (see  Fig.  35)  at  least  two  to  two  and 

one  half  diameters  in  length,  the 
jet  issues  full  bore  or  without 
contraction  at  the  point  of  dis- 
charge. 

If  A  be  the  sectional  area  of 
the  mouthpiece,  h  the  depth  of 
its  axis  below  the  water-surface, 
and  Q  the  amount  of  the  dis- 
charge. Then  experiment  shows 
that 

Q  =  .S2A  |/^.       .     (i) 

The  coefficient  .82  is  the  pro- 
duct of  the  coefficients  of  veloc- 
ity and  contraction,  but  the  co- 
efficient of  contraction  is  unity, 


FIG  35. 


and  therefore  the  coefficient  of  velocity  is  .82.  Now  the 
mean  coefficient  of  velocity  in  the  case  of  a  simple  sharp- 
edged  orifice  is  .947,  and  the  difference  between  .947  and  .82 
cannot  be  wholly  accounted  for  by  frictional  resistances,  but 
is  in  part  due  to  a  loss  of  head.  In  fact,  the  water  as  it  clears 
the  inner  edge  of  the  mouthpiece  converges  to  a  minimum  sec- 
tion MN  of  area  a  and  then  swells  out  until  at  M' N'  it  again 
fills  the  mouthpiece. 

Energy  is  wasted  in  eddy  motions  between  MN  and  M'N', 
where  the  action  is  similar  to  that  which  occurs  at  an  abrupt 
change  of  section. 

Let  /,  v  be  the  intensity  of  the  pressure  and  the  mean 
velocity  of  flow  at  the  point  of  discharge. 

Let  />, ,  v1  be  similar  symbols  for  the  contracted  section  MN. 

Let  />„  be  the  intensity  of  the  atmospheric  pressure. 


Remembering  that 


is  the  loss  of   head  "due  to 


shock  "  between  MN  and  M'N,  then  by  Bernoulli's  theorem 

•      •      (2) 


,    A  =  A      v    =  /_,   v 

•"  W   ~     W     '     2g        W*2f 


2g 


FLO  W   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  4! 

Hence 


W  2g 

and 


(I-)'! 


where  4.  =  coefficient  of  contraction  =  —  =  — .     Therefore 


an  equation  giving  the  velocity  of  flow  at  the  point  of  discharge. 
If  the  discharge  is  into  the  atmosphere,^  =/  and  equation 
4  becomes 

*  ' 


where 


/         \. 

1--IJ 


(5) 


^.=-+(7,-.)' 


If  cc  =  .62,  then  cv  =  .85,  while  experiment  gives  .82  as 
the  value  of  cv.  The  small  difference  between  .85  and  .82  is 
probably  due  to  frictional  resistance.  The  value  .82  for  cv 
makes  cc  approximately  .617. 

Again,  the  discharge  from  a  simple  sharp-edged  orifice  of 
same  sectional  area  as  the  mouthpiece  is  .62A  V2gh,  or  more 
than  24  per  cent  less  than  the  discharge  from  the  cylindrical 
mouthpiece.. 


42  HYDRA  ULICS. 

The  loss  of  head  between  MN  and  M'N' 


(by  eqs.  5  and  6) 


=  h(i  —  O  =  h  X  .3276 
=  —  ,  approximately. 

Thus  the  effective  head  is  only  \h,  instead  of  h. 

By  eq.  3  the   difference   between   the    pressure-heads   at 
MN  and  at  the  point  of  discharge 


=  — . h  =  //  -    — 


w 


=  £^,  very  nearly. 

Now  if  one  end  of  a  tube  is  inserted  in  the  mouthpiece  at 
the  contracted  section  (Fig.  35)  and  the  other  end  immersed 
in  a  vessel  of  water,  the  water  will  at  once  rise  to  a  height  /z,  in 

the  tube,  showing  that  the 
pressure  at  the  contracted 
section  is  less  than  that  due  to 
the  atmosphere.  By  careful 
measurement  it  is  found  that 
h^  is  very  nearly  equal  to  \hy 
which  verifies  the  theory. 

(d]  Divergent  Mouthpiece. 
— Suppose  that  for  the  cylin- 
drical mouthpiece  in  (c)  there 
I  is     substituted     a     divergent 

FIG-  36.  mouthpiece  of  the  exact  form 

of  the  issuing  jet  (see  Fig.  36),     Then — 


FLO  W   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  43 

(1)  The  mouthpiece  will  run  full  bore. 

(2)  There  will  be  no  loss  of  head  between  the  minimum 
section  MN  and  the  plane  of  discharge  AB,  as  there  is  now  no 
abrupt  change  of  section. 

Hence  by  Bernoulli's  theorem,   and    retaining   the   same 
symbols  as  in  (c), 


=+      =     +  (.) 

W  W        2g        W        2g 

If  the  discharge  is  into  the  atmosphere,/  =  /0,  and  therefore 

v*  =  2gh\      ....   '.     .     .     .     (2) 

or   introducing   a   coefficient  cv  (=  .98,  nearly,  for  a  smooth 
well-formed  mouthpiece), 


and  the  discharge  is 

.    ,    ;,  ._.,    .    (4) 


From  the  last  equation  it  would  appear  as  if  the  discharge 
would  increase  indefinitely  with  A,  but  this  is  manifestly 
impossible. 

In  fact,  by  eq.  I,  the  flow  being  into  the  air,  and  taking 


,  (c) 

W        W        2g\V* 


since  av^  =  Av.     But/,  cannot  be  negative,  and  therefore 


so  that 


a       '\  wk+l (7) 

gives  a  maximum  limit  for  the  ratio  of  A  to  a. 


44 


HYDRA  ULICS. 


— 


Now  —  =  34  feet  very  nearly,  and  the  last  equation  may  be 


written 


By  eqs.  4  and  7, 


(9) 


which  is  also  the  expression  for  the  discharge  through  the 
minimum  section  a  into  a  vacuum. 

If,  however,  the  sectional  areas  of  the  mouthpiece  at  the 
point  of  discharge  and  at  the  throat  are  in  the  ratio  of  A  to  a, 
as  given  by  eq.  7,  it  is  found  that  the  full-bore  flow  will  be  in- 
terrupted either  by  the  disengagement  of  air,  or  by  any  slight 
disturbance,  as,  for  example,  a  slight  blow  on  the  mouthpiece, 
and  hence,  in  practice,  it  is  usual  to  make  the  ratio  of  A  to  a 
sensibly  less  than  that  given  by  eq.  7. 

(e)  Convergent  Mouthpiece.  —  With  a  convergent  mouth- 
piece (Fig.  37)  two  points  are  to  be  noted  : 

(i)  There  is  a  contraction  within  the  mouthpiece,  followed 
by  a  swelling  out  of  the  jet  until  it  again  fills  the  mouthpiece. 


FIG.  37. 

Thus,  as  in  the    case  of  cylindrical  mouthpieces,  there  is  a 
"  loss  of  head  "  between  the  contracted  section  and  the  point 


FLO W   THROUGH  ORIFICES,   OVER   WEIRS,  ETC. 


45 


of  discharge,  and  also  a  consequent  diminution  in  the  velocity 
of  discharge. 

(2)  There  is  a  second  contraction  outside  the  mouthpiece 
due  to  the  convergence  of  the  fluid  filaments.  The  mean 
velocity  of  flow  (V)  across  the  section  is 


v'  =  C,' 


Cvr  being  the  coefficient  of  velocity  and  h  the  effective  head 
above  the  centre  of  the  section. 
Also,  the  area  of  this  section 

=  CC'X  area  of  mouthpiece  at  point  of  discharge 
=  CC'.A, 

Cc  being  the  coefficient  of  contraction.     Hence  the  discharge 
Q  is  given  by 


Q  =  CvfCc'A 


=  C'A 


C'(=  Cv'Ccf)  being  the  coefficient  of  discharge. 

The  coefficients  Cv'  and  Ce'  depend  upon  the  angle  of  con- 
vergence, and  Castel  found  that  a  convergence  of  13°  24'  gave 
a  maximum  discharge  through  a  mouthpiece  2-6  diameters  in 
length,  the  smallest  diameter  being  .05085  foot. 


TABLE   GIVING   CASTEL'S    RESULTS. 


Angles  of 
Convergence. 

Cc 

<v 

C' 

Angles  of 
Convergence. 

<v 

Cv 

C' 

o°     o' 

•999 

.830 

.829 

13  °24' 

.983 

.962 

.946 

I      36 

i  .000 

.866 

.866 

14     28 

•979 

.966 

.941 

3     10 

1.  001 

.894 

•895 

16     36 

.969 

.971 

.938 

4    10 

i.  002 

.910 

.912 

19     28 

•953 

.970 

.924 

5     26 

1.004 

.920 

.924 

21        0 

•945 

.971 

.918 

7    52 

.998 

•931 

.929 

23      o 

•937 

•974 

•913 

8     58 

.992 

.942 

•934 

29      58 

.919 

•975 

.896 

10      20 

.987 

•950 

.938 

40      20 

.887 

.980 

.869 

12        4 

.986 

•  955 

.942 

48      50 

.661 

.984 

.847 

46 


HYDRAULICS. 


18.  Radiating  Current. — As  an  application  of  Bernouilli's 
theorem,  consider  the  steady  plane  motion  of  a  body  of  water 
flowing  radially  between  two  horizontal  planes  a  ft.  apart  and 
symmetrical  with  respect  to  a  central  axis  (Fig.  38). 

Let  v  ft.  per  second  be  the  velocity  at  the  surface  of  a  cyl- 

FIG.  38. 


FIG.  39- 

inder  of  radius  r  ft.  described  about  the  same  axis.     Then  the 
volume  Q  crossing  the  second  per  surface  is 

Q  =  2nr  .  av, 
and  therefore 


Q 
rv  =    =£-  =  a  constant, 


since  Q  is  constant. 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  4? 

Thus  v  increases  as  r  diminishes,  and  becomes  infinitely 
great  at  the  axis;  but  it  is  evident  that  the  current  must  take 
a  new  course  at  some  finite  distance  from  the  axis. 

If  p  is  the  pressure  at  any  point  of  the  cylindrical  surface 
3  ft.  above  datum,  then,  by  Bernoulli's  theorem, 

z  +  —  +  —  =  a  constant  =  h  =  y  +  — , 

denoting  the  dynamic  head  z  +  —  by  y.     Hence 

w 

,  v*  Q*  a  constant 


2g       Zn'a'r'g  r' 

and  therefore 

r*(h  —  y)  =  a  constant 

is  an  equation  giving  the  free  surfaces  of  the  pressure  columns 
(Fig.  39).  These  surfaces  are  thus  generated  by  the  revolution 
of  Barlow's  curve. 

The  surfaces  of  equal  pressure  are  also  given  by  an  equa- 
tion of  the  same  form. 

19.  Vortex  Motion. — A  vortex  is  a  mass  of  rotating  fluid, 
and  the  vortex  is  termed  free  when  the  motion  is  produced 
naturally  and  under  the  action  of  the  forces  of  weight  and 
pressure  only. 

In  the  radiating  current  already  discussed,  assume  that  the 
direction  of  motion  at  each  point  is  turned  through  a  right 
angle. so  that  the  mass  of  water  will  now  revolve  in  circular 
layers  about  the  central  axis.  Also,  if  there  is  a  slow  radial 
movement,  so  that  fluid  particles  travel  from  one  circular  stream- 
line to  another,  it  is  assumed  that  these  particles  freely  take  the 
velocities  proper  to  the  stream-lines  which  they  join.  Such  a 
motion  is  termed  a  free  circular  vortex. 

The  motion  being  steady  and  horizontal,  the  equation 

z  +  —  -| =  a  constant  =  //,    .     .     .     .     (i) 


48  HYDRA  ULICS. 

holds  good  at  every  point  of  a  circular  stream  of  radius  r. 
Again, 

w.d\z-\-— )  =  increment  of  dynamic  pressure  between  two- 
consecutive  elementary  stream-lines 

=  deviating  force 

=  centrifugal  force  of  an  element  between  the 
two  stream-lines 


Hence 

,/         p\  w    ,        wv1     j 

w  .  d\z  4-  —  I  =  —  —vdv  =  --  .  dry 
\        wi  g  gr 

and  therefore 


so  that  vr  —  a  constant,  and  v  varies  inversely  as  r,  as  in  the 
case  of  the  radiating  current.  Therefore  the  curves  of  equal 
pressure  will  also  be  the  same  as  in  a  radiating  current. 

Free  Spiral  Vortex.  —  Suppose  that  the  motion  of  a  mass 
of  water  with  respect  to  an  axis  O  is  of  such  a  character  that  at 
any  point  J/the  components  of  the  velocity  in  the  direction  of 
OM,  and  perpendicular  to  OM,  are  each  inversely  proportional 
to  the  distance  OM  from  O.  The  motion  is  thus  equivalent  to 
the  superposition  of  the  motions  in  a  radiating  current  and  in 
a  free  circular  vortex  ;  and  if  0  is  the  angle  between  <9J/and  the 
direction  of  the  stream-line  at  M,  v  cos  6  and  v  sin  9  are  each 
inversely  proportional  to  OM,  and  therefore  6  must  be  con- 
stant. Hence  the  stream-lines  must  be  equiangular  spirals  and 
the  motion  is  termed  a  free  spiral  vortex. 

This  result  is  of  value  in  the  discussion  of  certain  turbines 
and  centrifugal  pumps.  A  steady  free  surface  in  the  case  of  a 


*FLOW    THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  49 

free  spiral  vortex  is  impossible,  as  the  stream-lines  cross  the 
surfaces  of  equal  pressure,  which  are  the  same  as  before. 

Also  if  /„,  z/0,  r0  are  the  pressure,  radius,  and  velocity  at  any 
other  point  at  the  same  elevation  2  above  datum,  then 


W         2g  W         2g 

and  the  increase  of  pressure-head 


w  2g          2gr*  2g 

Forced  Vortex.  —  A  forced  vortex  is  one  in  which  the  law 
of  motion  is  different  from  that  in  a  free  vortex.  The  simplest 
and  most  useful  case  is  that  in  which  all  the  particles  have  an 
equal  angular  velocity,  so  that  the  water  will  revolve  bodily,  the 
velocity  at  any  point  being  directly  proportional  to  the  distance 
from  the  axis. 

As  before, 


wl       g  r 
But 

v  oc  r  =  cor, 

co  being  the  constant  angular  velocity  of  the   rotating  mass. 
Therefore 

p\  GO* 

-- 


_7I  I      r    \  ^JtJ  J 

d  U  +  --1  =  — r  .  dr. 
\         wl       g 

Integrating, 

z  +  —  =  -    -  +  a  constant  =  -  f  +  a  constant. 

Hence,  if/0,  r0,  v0  are  the  pressure,  radius,  and  velocity  for 
any  second  point  at  the  same  elevation  z  above  datum,  then 

W  2v  2<>  °' 

<5  <5 


HYDRA  ULICS. 


If  the  second  point  is  on  the  axis  of  revolution,  then  ra  =  o, 
and  the  last  equation  becomes 


W  2P~ 

1  hus  the  free  surface  of  the  pressure  columns  is  evidently  a 
paraboloid  of  revolution  with  its  vertex 
at  O,  as  in  Fig.  40. 

A  compound  vortex  is  produced  by 
the  combination  of  a  central  forced  vor- 
tex with  a  free  circular  vortex,  the  free 
surface  being  formed  by  the  revolution 
of  a  Barlow  curve  and  a  parabola. 

For    example,   the   fan    of   a   centri- 
fugal pump  draws  the  water  into  a  forced 
FIG.  40.  vortex  and  delivers  it  as  a  free  spiral  vor- 

tex into  a  whirlpool-chamber  (Chap.  VII.). 

In  this  chamber  there  is  thus  a  gain  of  pressure-head,  and 
the  water  is  therefore  enabled  to  rise  to  a  corresponding  addi- 
tional height.  James  Thomson  adopted  the  theory  of  the  corn- 
compound  vortex  as  the  principle  of  the  action  of  his  vortex 
turbine. 

20.  Large  Orifices  in  Vertical  Plane  Surfaces. — The 
issuing  jet  is  approximately  of  the  same  sectional  form  as 
the  orifice,  and  the  fluid  filaments  converge  to  a  minimum 
section  as  in  the  case  of  simple  sharp-edged  orifices. 

(a)  Rectangular  Orifice  (Fig.  41). — Let  E,  F  be  the  upper  and 
lower  edges  of  a  large  rectangular  orifice  of  breadth  B,  and  let 
H^ ,  H^  be  the  depths  of  E  and  F,  respectively,  below  the  free 
surface  at  A.  If  u  be  the  velocity  with  which  the  water  reaches 

the  orifice,  then  H  =  --  is  the  fall  of  free  surface  which  must 

have  been  expended  in  producing  the  velocity  u. 

Hence,  Hl-\-  H  and  H^  +  H  are  the  true  depths  of  the 
edges  E  and  F  below  the  surface  of  still  water. 

Let  A/TV  be  the  minimum  or  contracted  section,  and  assume 
that  it  is  a  rectangle  of  breadth  b. 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  5 1 

Let  hl ,  h^  be  the  depths  of  M  and  N,  respectively,  below 

the  free  surface  at  A. 
Then  hl-\-  H,  h^  -\-  H  are  the  true  depths  of  M  and    N 

below  the  surface  of  still  water. 

First,  let  the  flow  be  into  the  air,  the  orifice  being  clear 
above  the  tail-water  level. 

Consider  a  lamina  of  the  fluid  at  the  section  MN  of  the 


1 

j 

? 

;-     "' 

"==:iEi5= 

4 

T 
1 
1 

!    i- 

f  : 

~—  f=£ 

1 

i 

J 

1          1 

f  ' 

r       i 

E 

f 
| 

i 

V| 

f1 

» 
. 

2 

. 

M-1--  ;— 

H 

»L.-i__ 

J 

L..[- 

i 

i 

1 

FIG.  41. 

width  of  the  section  and   between  the  depths  x  and  ^r  +  dx 
below  the  surface  of  still  water. 

The  elementary  discharge  dq,  in  this  lamina,  is 

dq  =  bdx  <y/2gx, 
and  therefore  the  total  discharge  Q  across  the  section  MNis 


/[*h.i  +  H 
dq=        b.dxjlgx 
J  hi  -^H 


Put*= 
Then 


t\.  .  .  (i) 


HYDRA  ULICS. 


The  coefficient  c  is  by  no  means  constant,  but  is  found  to 
vary  both  with  the  head  of  water  and  also  with  the  dimensions 
of  the  orifice,  and  can  only  be  determined  by  experiment. 
Second,  let  the  orifice  be  partially  (Fig.  42)  submerged,  and 
and  let  //,  be  the  depth  between  the 
surface  of  the  tail-race  water  and  the 
free  surface  at  A. 

By  what  precedes,  the  discharge  Qt 
through  EG,  the  portion  of  the  orifice 
clear  above  the  tail-race,  is 


.  (2) 

Every  fluid  filament  flows  through 
the  portion  GF  of  the  orifice  under  an 
effective  head  Hz  -f-  H,  and  therefore 
with  a  velocity  equal  to 


FIG.  42. 


Hence  the  discharge  <23  through  GF  is 


...     (3) 

and  the  total  discharge  Q  is  equal  to  Q,  +  Qt. 

The  coefficients  clt  c^  are  to  be  determined  by  experiment, 

and  if  cl  =  c^  =  c, 


.     (4) 


Third,  let  the  orifice  be  wholly  submerged  (Fig.  43).     Then 
the  total  discharge  Q  is  evidently 


Q  = 


'.  +  Jf,      •    •     •     (5) 


c  being  a  coefficient  to  be  determined  by  experiment. 

If  the  velocity  of  approach,  «,  is  sufficiently  small   to   be 


--  a 


FLOW    THROUGH  ORIFICES,   OVER   WEIRS,  ETC. 


53 


disregarded  without  sensible  error,  then  H  =  o,  and  equations 
i,  4,  and  5,  respectively,  become 


(8) 


(b)  Circular  Orifices. — Let  Fig.  44 
represent  the  minimum  section  of 
the  circular  jet  issuing  from  a  circu- 
lar orifice. 

Let  26  be  the  angle  subtended  at  the  centre  by  the  fluid 


FIG.  43- 


FIG.  44. 

lamina  between  the  depths  x  and  x  +  dx  below  the  surface 
of  still  water. 

Let  r  be  the  radius  of  the  section  so  that  2r  =  h^  —  h^  h^ 
and  h^  being,  as  in  (a),  the  depths  of  the  highest  and  lowest 
points  of  the  orifice  below  the  free  surface  at  A. 

H,  as  before,  is  the  head  corresponding  to  the  velocity  of 
approach. 


54  HYDRA  ULICS. 

Then  the  area  of  the  lamina  under  consideration 

=  2r  sin  9  .  dx, 

and  the  elementary  discharge,  dq,  in  this  lamina,  is 
dq  =  2r  sin  6.  dx^2gx* 

h.+H+k.+H 
But*=-        -^      - 

and  therefore 

dx 
Hence 


dq  =  2r'sin'fttV'         *2  -rcos 

and  the  total  discharge  Q  is 


-  r  cos  ff  V       (9) 


21.  Notches  and  Weirs.  —  When  an  orifice  extends  up  to 
the  free-surface  level  it  becomes  what  is  called  a  notch. 

A  weir  is  a  structure  over  which  the  water  flows,  the 
discharge  being  in  the  same  conditions  as  for  a  notch. 

Rectangular  Notch  or  Weir.  —  The  discharge  may  be  found 
by  putting  Hl  =  o. 

Thus  equation  I  becomes 


(10) 


If  the  velocity  of  approach  be  disregarded,  then  H  •=.  o, 
and  the  last  equation  becomes 


(ii) 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  55 

and  //,  is  the  depth  to  the  bottom  of  the  notch  or  to  the  crest 
of  the  weir. 

The  effective  sectional  area  of  the  water  flowing  through  a 
rectangular  notch,  or  over  a  weir,  is  less  than  BH.t,  because  of 
(a)  crest  contraction,  (b)  end  contraction,  (c)  the  fall  of  the  free 
surface  towards  the  point  of  discharge. 

It  is  reasonable  to  assume  that  the  diminution  of  the  actual 
sectional  area,  BH^  ,  due  to  crest  contraction  and  to  the  fall  of 
the  free-surface  level  is  proportional  to  the  width  B  of  the 
opening,  and  that  the  effect  of  end  contractions  is  very  nearly 
the  same  both  for  wide  and  narrow  openings. 

Francis,  in  his  Lowell  weir  experiments,  found  that  for 
depths  H^-\-  H  over  the  crest,  varying  from  3  in.  to  24  in., 
and  for  widths  B  not  less  than  three  times  the  depth,  a  per- 
fect end  contraction  had  the  effect  of  diminishing  the  width  of 
the  fluid  section  by  an  amount  approximately  equal  to  one 

f-f  —I—  J-T 
tenth  of  the  depth,  or      2  "*"  —  ,    so   that    the    effective    width 


IO 

Thus,  if  there  are  n  end  contractions,  the  effective  width 
=  B  —  —  (//,  +  H),  and  the  equation  giving  the  discharge 
becomes 


Q  =  -c  \  B  -  £(#.  +/0  }  S&\(Ht  +  H)*-  H*\.     (12) 


According  to  Francis,  the  average  value  of  c  in  this  equa- 
tion is  .622. 

Circtilar  Notch. — In  equation  9,  Art.  20,  put  h^  =  o  and 
h  =  2r.  Then 


Tsin2 


Q  =  2r2  ^2g       sin2  BH  +  2r  sin' 


56  HYDRAULICS. 

and  if  the  velocity  of  approach  be  disregarded,  so  that  H  =  o, 


C*  6 

Q  =  2r*  -v/4£-  /    sin2  6  sin  -  .  dB 

*/  o 


2 


sn     -sin  sn 


(13) 


22.  Triangular  Notch, — Disregard    the    velocity   of    ap- 
proach and   let  B  be  the  width  of  the  free  surface. 

As  before,    consider    a    lamina 
of  fluid  between  the  depths  x  and 

^  ^  B 

The  area  of  the  lamina  =  -77- 

\  —  x}dx,  and  the  discharge  in 
this  lamina  is 


7} 

dq  =  C(H*  —  x 


Hence  the  total  discharge  Q  is 

r"*   ry 


" 


(14) 


c  is  a  coefficient  introduced  to  allow  for  contraction,  etc.,  and 
Professor  James  Thomson  gives  .617  as  its  mean  value  for  a 
sharp-edged  triangular  notch. 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  $? 

r> 

Now  the  ratio  jy  is  constant  in  a  triangular  notch  and  varies 

**\ 
in  a  rectangular  notch.      Hence  Thomson  inferred  and  proved 

by  experiment  that  the  value  of  c  is  more  constant  for  trian- 
gular than  for  rectangular  notches,  so  that  a  triangular  notch 
is  more  suitable  for  accurate  measurements. 

Example.  —  A  sharp-edged  triangular  notch  is  opened  in 
the  side  of  a  reservoir,  and  the  water  flows  out  until  the  free- 
surface  level  sinks  to  the  bottom  of  the  notch. 

The  discharge  in  the  short  interval  dt,  when  the  depth  of 
water  in  the  notch  is  x  ft., 


=  —cmx 


mx  being  the  width  of  the  free  surface  corresponding  to  the 
depth  x,  and  m  a  coefficient  depending  upon  the  angle  of  the 
notch. 

Again,  S  .  dx  is  the  quantity  of  the  water  which  leaves  the 
reservoir  in  the  same  time  dt,  S  being  the  horizontal  sectional 
area  of  the  reservoir  corresponding  to  the  depth  x.  Hence 


4   ,/— 
and  therefore 


—  \/2gcmxldt  =  —  Sdx, 


—  \f2gcmdt  =  —  Sx~*dx, 

so  that  the  time  in  which  the  free  surface  sinks  to  the  required 
level 

x 


15     c 

=  --  7^—  / 
4  V2gcmJQ 

X  being  the  initial  depth. 


58  HYDRAULICS. 

If  5  is  constant,  then 

the  time  = 


23.  Broad-crested  Weir.  —  Let  Fig.  46  represent  a  stream 
flowing  over  a  broad-crested  weir.     On  the  up-stream  side  the 


FIG.  46. 

free  surface  falls  from  A  to  B.  For  a  distance  BD  on  the  crest 
the  fluid  filaments  are  sensibly  rectilinear  and  parallel;  the 
inner  edge  of  the  crest  is  rounded  so  as  to  prevent  crest  con- 
traction. 

Consider  a  filament  ab,  the  point  a  being  taken  in  a  part  of 
the  stream  where  the  velocity  of  flow  is  so  small  that  it  may  be 
disregarded  without  sensible  error. 

Let  A  be  the  thickness  MN  of  the  stream  at  b. 

Let  the  horizontal  plane  through  N  be  the  datum  plane. 

Let  #„  z  be  the  depths  below  the  free  surface  of  a  and  b. 

Let  hl  be  the  elevation  of  a  above  datum. 

Let/0,  /,,  p  be  the  atmospheric  pressure  and  the  pressures 
at  a  and  b. 

Let  v  be  the  velocity  of  flow  at  b. 

Then,  by  Bernoulli's  theorem, 


W 


W 


2g 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  59 

But 


=  *i  +  and          =  ,  +       ; 

W  WWW 


therefore 


and  hence 


_  =  k,  +  zl  -  A  =  H,  -  A, 


//",  being  the  depth  of  the  crest  of  the  weir  below  the  surface 
of  still  water. 

Thus,  if  B  be  the  width  of  the  weir,  the  discharge  Q  is 


(16) 


From  this  equation  it  appears  that  Q  is  nil  both  when 
A  =  o  and  when  A  =  //,.  Hence  there  must  be  some  value 
of  A  between  o  and  //,  for  which  Q  is  a  maximum.  This  value 
may  be  found  by  putting 


and  the  expression  for  the  discharge  becomes 

,  =  .3855  V^rf,    .  '  .    (17) 


which  is  the  maximum  discharge  for  the  given  conditions. 

Experiment  shows  that  the  more  correct  value  for  the  dis- 
charge is 

.  .    .    (18) 


60  HYDRAULICS. 

This  formula  agrees  with  the  ordinary  expression  for  the 
discharge  over  a  weir  as  given  by  equation  u,  if  c  =  .525. 

It  might  be  inferred  that  for  broad-crested  weirs  and  large 
masonry  sluice-openings  the  discharge  should  be  determined 
by  means  of  equation  18  rather  than  by  the  ordinary  weir 
formula,  viz.,  equation  n. 

It  must  be  remembered,  however,  that  in  deducing  equa- 
tion 17  frictional  resistances  have  been  disregarded  and  the 
gratuitous  assumption  has  been  made  that  the  stream  adjusts 
itself  to  a  thickness  /  which  will  give  a  maximum  discharge. 
The  theory  is  therefore  incomplete. 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  6 1 


EXAMPLES. 

I.  A  frictionless  pipe  gradually  contracts  from  a  6-in.  diameter  at  A 
to  a  3-in.  diameter  at  B,  the  rise  from  A  to  £  being  2  ft.  If  the  de- 
livery  is  i  cubic  foot  per  second,  find  the  difference  of  pressure  between 
the  two  points  A  and  B.  Ans.  500  Ibs.  per  sq.  ft. 

.      2.  In  a  frictionless  horizontal  pipe  discharging  10  cubic  feet.of  water 
per  second,  the  diameter  gradually  changes  from  4  in.  at  a  point  A  to       i/C 
6  in.  at  a  point  B.     The  pressure  at  the  point  ^5*is  100  Ibs.  per  square 
inch  ;  find  the  pressure  at  the  point  A.  Ans.  4118  Ibs.  per  sq.  ft. 

3.  A  ^-in.  horizontal  pipe  is  gradually  reduced  in  diameter  to  -J-  in. 
and  then  gradually  expanded  again  to  its  mouth,  where  it  is  open  to  the 
atmosphere.     Determine  the  maximum  quantity  of  water  which  can  be 
forced  through  the  pipe  (a)  when  the  diameter  of  the  mouth  is  \  in.,  (b) 
when  the  diameter  is  f  in.     Also  determine  the  corresponding  velocities 
at  the  throat  and  the  total  heads  (neglect  friction,  which,  however,  is 
very  considerable).      Ans.  (a)  .24  cub.  ft.  per  min.;  46.7  ft.  per.  sec. 

(b)  .239  cub.  ft.  per.  min.;  46.66  ft.  per  sec. 

4.  A  short  horizontal  pipe  A BC  connecting  two  reservoirs  gradually 
contracts  in  diameter  from  i  inch  at  A  to  £  inch  at  B  and  then  enlarges 
to  i  inch  again  at  C.     If  the  height  of  the  water  in  the  reservoir  over  C 
be  12  inches,  determine  the  maximum  flow  through  the  pipe  and  sketch 
the  curve  of  pressures.     Also  obtain  an  equation  for  this  curve,  assum- 
ing the  rates  of  contraction  and  expansion  of  the  pipe  to  be  equal  and 
uniform.  Ans.  3.75  cub.  ft.  per  min. 

5.  The  pipe  DE  in  the  figure  is  gradually  contracted  in  diameter 
from  D   to  E,  where   it   is   enclosed   in 

another   pipe  ABC,  expanding   from  B 

towards  A  and  C\  at  C  it  is  open  to  the 

atmosphere  and  at  A  it  is  connected  with  D/ 

a  reservoir  R  ;  the  water  surface    in  R 

being  h'  below  the  horizontal  axis  of  DE. 

If  the  velocity  in  DE  at  E  be  v  and  the 

velocity  in  AB  at  B  be  F,  what  will  be  the 

common  velocity  after  uniting?    Explain 

what  becomes  of  the  energy  lost  in  im-  ^ 

pact.     If  the  diameters  at  E,  B,  arid  C 

are  \  in.,  f  in.,  and  i  in.,  the  distance  between  the  outside  of  E  and  inside 


62  HYDRA  ULICS. 

of  B  being  T\  inch,  find  the  ratio  of  the  quantity  pumped  from  R  to  the 
flow  through  DE. 

6.  A  3-in.  pipe  gradually  expands  to  a  bell-mouth  ;  if  the  total  head, 
//,  be  40  ft.,  find  the  greatest  diameter  of  the  mouth  at  which  it  will 
run  full  when  open  to  the  atmosphere.     Compare  the  discharge  from 
this  pipe  with  the  discharge  when  the  pipe  is  not  expanded  at  the  mouth. 

Ans.  4.8  in.;  discharge  is  18.63  cub.  ft-  Per  minute  with  bell- 
mouth  and  7.337  cub.  ft.  per  minute  without  bell-mouth. 

7.  The  pressure  in  a   12-  in.  pipe  at  A  is  50  Ibs.;  the  pipe  then  en-  ^ 
larges  to  a  i5-in.  pipe  at  B,  the  rise  from  A  to  B  being  3  ft.;  the  dis-  / 
charge  is  Q  cubic  feet  per  minute.    Find  the  pressure  at^;  also  find  the 
pressure  at  a  point  C,  the  rise  from  B  to  C  being  6  ft, 


(6637.5  +  T£ 


Ans.    6637.5  +  lbs'     er  sc-  ft' 


8.  Two  equal  pipes  lead,  one  from  the  steam-space,  the  other  from 
the  water-space  of  a  boiler  at  pressure/;  Ss  is  the  density  of  the  steam 
and  Sw  that  of  the  water.     Assuming  Torricelli's  theory  to  hold  for  rate 
of  efflux  of  steam  and  water,  show  that 

vel.  of  steam-jet  _  */•£»  _  quantity  of  water-jet  _  energy  of  steam-jet 
vel.  of  water-jet  ~~  *    Ss  ~  quantity  of  steam-jet  ~~  energy  of  water-jet, 

and  that  the  momentum  of  each  jet  is  the  same. 

9.  Find   the   head   required   to   give   i   cub.  ft.  of  water  per  second 
through  an  orifice  of  2  square  inches  area,  the  coefficient  of  discharge     [/ 
being  .625.     (g  =  32.)  Ans.  206  ft. 

10.  The  area  of  an  orifice  in  a/fcnin  plate  was  36.3  square  centimetres, 
the  discharge  under  a  head  0^/^396  metres  was  found  to  be  .01825  cubic 
metre  per  second,  and  the  velocity  of  flow  at  the  contracted  section,  as  \^/ 
determined  by  measurements  of  the  axis  of  the  jet,  was  7.98  metres  per 
second.     Find  the  coefficients  of  velocity,  contraction,  discharge,  and  re- 
sistance.    (^  =  9.81.)  Ans.  .978;  .631;  .617;  .045. 

11.  The  piston  of  a  12-in.  cylinder  containing  saltwater  is  pressed 
down  under  a  force  of  3000  lbs.     Find   the  velocity  of  efflux  and  the    \J 
volume  of  discharge  at  the  end  of  the  cylinder  through  a  well-rounded 

i  -in.  orifice.     Also  find  the  power  exerted. 

Ans.  60.373  ft.  per  sec.;  .1691  cub.  ft.  per  sec.;  1.166  H.  P. 

12.  In  the  condenser  of  a  marine  engine  there  is  a  vacuum  of  26^  in. 
of  mercury  ;  the  injection  orifices  are  6  ft.  below  the  sea-level.     With 
what  velocity  will  the  injection-water  enter  the  condenser?  (Neglect  re- 
sistance.) Ans.  25.3  ft.  per  sec. 

13.  Water  in  the  feed-pipe  of  a  steam-engine  stands  12  ft.  above  the 


FLO W    THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  63 

surface  of  the  water  in  the  boiler  ;  the  pressure  per  sq.  in.  of  the  steam  is 
20  Ibs.,  of  the  atmosphere  15  Ibs.  Find  the  velocity  with  which  the 
water  enters  the  boiler.  Ans.  5.376  ft.  per  sec. 

14.  The  injection  orifice  of  a  jet  condenser  is  5  ft.  below  sea-level 
and  vacuum  =  27  in.  of  mercury.     Find  velocity  of  water  entering  con- 
denser, supposing  three  fourths  of  the  head  lost  by  frictional  resistance. 

Ans.  23.86  ft.  per  sec. 

15.  A  vessel  containing  water  is  placed  on  scales  and  weighed.    How 
will  the  weight  be  affected  by  opening  a  small  orifice  in  the  bottom  of 
the  vessel  ? 

1 6.  Water  is  supplied  by  a  scoop  to  a  locomotive  tender  at  7  ft.  above 
trough.     Find  lowest  speed  of  train  at  which  the  operation  is  possible. 

Ans.  14.44  miles  per  hour. 

Also  find  the  velocity  of  delivery  when  train  travels  at  40  miles  per 
hour,  assuming  half  the  head  lost  by  frictional  resistance. 

Ans.  35.68  ft.  per.  sec. 

17.  The  head  in  a  prismatic  vessel  at  the  instant  of  opening  an  orifice 
was  6  ft.  and  at  closing  it  had  decreased  to  5  ft.     Determine  the  mean 
constant  head  h  at  which,  in  the  same  time,  the  orifice  would  discharge 
the  same  volume  of  water.  Ans.  5.434  ft. 

18.  A  prismatic  vessel  5.747  in.  in  diameter  has   an  orifice  of  .2  in. 
diam.  at  the  bottom;    the   surface   sinks  from  16  in.  to  12  in.   in   53 
seconds.     Find  the  coefficient  of  discharge.  Ans.  .6. 

19.  A  prismatic  basin  with  a  horizontal  sectional  area  of  9  sq.  ft.  has 
an  orifice  of  .09  sq.  ft.  at  the  bottom  ;  it  is  filled  to  a  depth  of  6  ft.  above 
the  centre  of  the  orifice.     Find  the  time  required  for  the  surface  to  sink 
2  ft.,  3^  ft.,  5  ft.  Ans.  260  sec.;  502  sec.;  838  sec. 

20.  The  water  in  a  cylindrical  cistern  of  144  sq.  in.  sectional  area  is 
16  ft.  deep.     Upon  opening  an  orifice  of  I  sq.  in.  in  the  bottom  the 
water  fell  7  ft.  in  i  minute.     Find  the  coefficient  of  discharge.     The  co- 
efficient of  contraction  being  .625,  find  the  coefficients  of  velocity  and 
resistance.  Ans.  .6  ;  .96  ;  0.85. 

21.  How  long  will  it  take  to  fill  a  paraboloidal  vessel  up  to  the  level 
of  the  outside  surface  through  a  hole  in  the  bottom  2  ft.  under  water? 
(g  =  32  and  c  =  .625.) 

1 76  |/2  B 
Ans.  —j,  B  being  the  parameter  of  the  parabola  and  A  the 

sectional  area  of  the  orifice. 

22.  How  long  will  it  take  to  fill  a  spherical  Vessel  of  radius  r  up  to 
the  level  of  the  outside  surface  through  a  hole  of  area  A  at  bottom  2  ft. 
under  water  ?     (g  =  32  and  c  =  .625.) 

Ans' 


64  HYDRAULICS. 

23.  A  vessel  full  of  water  weighs  350  Ibs.  and  is  raised  vertically  by 
means  of  a  weight  of  450  Ibs.     Find  the  velocity  of  efflux  through  an 
orifice  in  the  bottom,  the  head  being  4  ft.  Ans.  17.02  ft.  per  sec. 

24.  A  vessel  full  of  water  makes  loo-revols.  per  min.    Find  the  velocity 
of  efflux  through  an  orifice   2  ft.  below  the  surface  of  the  water  at  the 
centre.  Ans.  33.4  ft.  per  sec. 

What  will  be  the  velocity  if  the  vessel  is  at  rest  ? 

Ans.  1 1. 35  ft.  per  sec. 

25.  The  jet  from  a  circular  sharp-edged  orifice,  £  in.  in  diameter,  un- 
der a  head  of  18  ft.,  strikes  a  point  distant  5  ft.  horizontally  and  4.665 
in.  vertically  from    the   orifice.      The   discharge    is   98.987   gallons    in 
569.218  seconds.    Find  the  coefficients  of  discharge,  velocity,  contraction, 
and  resistance.  Ans.  .6014;  .945;  .636;  .118. 

26.  A  square  box  2  ft.  in  length  and  i  ft.  across  a  diagonal  is  placed 
with  a  diagonal  vertical  and  filled  with  water.    How  long  will  it  take  for 
the  whole  of  the  water  to  flow  out  through  a  hole  at  the  bottom  of  .02 
sq.  ft.  area  ?     (c  —  .625.)  Ans.  97.48  sees. 

27.  A  pyramid  2  ft.  high,   on  a  square  base,  is  inverted  and  filled 
with  water.     Find  the  time  in  which  the  water  will  all  run  out  through 
a  hole  of  .02  sq.  ft.  at  the  apex.    A  side  of  the  base  is  i  ft.  in  length. 
(c.  —  .625.)  Ans.  15.08  sec, 

28.  Find  the  discharge  under  a  head  of  25  ft.  through  a  thin-lipped 
square  orifice  of  i  sq.  in.  sectional  area,  (a)  when  it  has  a  border  on  one 
side,  (b)  when  it  has  a  border  on  two  sides. 

Ans.  (a)  .3575  cu.  ft.  per  sec.;  (b)  .3706  cu.  ft.  per  sec. 

29.  A  vessel  in  the  form  of  a  paraboloid  of  revolution  has  a  depth  of 
16  in.  and  a  diam.  of  12  in.  at  the  top.     At  the  bottom  is  an  orifice  of 
i  sq.  in.  sectional  area.     If  water  flows  into  the  vessel  at  the  rate  of  2TV 
cubic  feet  per  minute,  to  what  level  will  the  water  ultimately  rise  ?     How 
long  will  it  take  to  rise  (a)  11  in.,  (b)  11.9  in.,  (c)   11.99  m->  (X)  I2  in- 
above   the  orifice?     If  the  supply  is  now  stopped,  how  long  (e)  will  it 
take  to  empty  the  vessel  ? 

Ans.  12  inches;    (a)  83.095  sec.;  (b)  124.2  sec.;    (c)  263.9  sec.; 
(d)  an  infinite  length  of  time  ;  (e)  11.3  sec. 

30.  If  the  vessel  in  Question  29  is  a  semi-sphere  i  ft.  in  diameter,  to 
what  height  will  the  water  rise  ?     How  long  will  it  take  for  the  water  to 
rise  (a)  11   in.,  (b)  12  in.  above  the  orifice  ?     How  long  (c)  will  it  take  to 
empty  the  vessel  ? 

Ans.  12  inches  ;  (a)  67.16  sec. ;  (£)  81.46  sec. ;  (c)  24.13  sec. 

31.  In  a  vortical  motion  two  circular  filaments  of  radii  ri ,  r2 ,  of  ve- 
locities Vi,Vt,  and  of  equal  weight  Ware  made  to  change  place.     Show 

7/2 

that  a  stable  vortex  is  produced  if  — =const.;  and  if  r2  >  r\ ,  show  that 
the  surfaces  of  equal  pressure  are  cones. 


FLOW   THROUGH1  ORIFICES,  OVER  WEIRS,  ETC.  65 

32.  Prove  that  for  a  Borda's  mouthpiece  running  full  the  coefficient 
of  discharge  is  — . 

4/2 

33.  The  surface  of  the  water  in  a  tank  is  kept  at  the  same  level; 
obtain  the  discharge  at  60  in.  below  the  surface  (a)  through  a  circular 
orifice  i  sq.  in.  in  area,  (U)  through  a  cylindrical  ajutage  of  the  same 
sectional  aYea  fitted  to  the  outside,  (c)  through  the  same  ajutage  fitted 
to  the  inside,  and  determine  the  mechanical  effect  of  the  efflux  in  each 


case.  Ans.  (a)  4  36    Ibs.  per  sec. 

(ff)  6.356   "       "      " 

(rf  3.488   ••       "      " 


20.514  ft.-lbs.  per  sec. 
21.369      "        "      " 

1744        "        "      " 


34.  Water  is  discharged  under  a  head  of  64  ft.  through  a  short  cylin- 
drical mouthpiece  12  in.  in  diameter.     Find   (a)   the  loss  of  head  due 
to  shock,  (£)  the  volume  of  disdharge  in  cubic  feet  per  secJnd,  (c)  the 
energy  of  the  issuing  jet.     (g  =  32.) 

Ans.  (a)  20.96  ft. ;  (8)  51.54  cub.  ft. ;  (c)  393.8  H.  P. 

35.  If  a  bell-mouth  is  substituted  for  the  mouthpiece  in  the  preced- 
ing question,  find  the  discharge  and  the  mechanical  effect  of  the  jet. 

Ans.  61.6  cub.  ft.  per  sec. ;  470.6  H.  P. 

36.  Compare  the  energies  of  a  jet  issuing  under  an  effective  head  of 
100  ft.  through  (i)  a  12-in.  cylindrical  ajutage,  (2)  a  12-in.  divergent  aju- 
tage, (3)  a  12-in.  convergent  ajutage,  the  angle  of  convergence  being  21°. 
Draw  the  plane  of  charge  in  each  case. 

Ans.  (i)  393.8  H.  P.;  (2)  672.28  H.  P.;  (3)  618.23  H.  P. 

37.  Find  the  discharge  through  a  rectangular  opening  36  in.  wide 
and   10  in.  deep  in  the  vertical  face  of  a  dam,  the  upper  edge  of  the 
opening  being  10  ft.  below  the  water  surface. 

Ans.  40.2  cub.  ft.  per  sec. 

38.  Find  the  discharge  in  pounds  per  minute  through  a  Borda's 
mouthpiece   i  in.   in  diameter,  the  lip  being   12    in.  below  the  water- 
surface.  Ans.  87.714  Ibs. 

39.  Sometimes  the  crest  of  a  dam  is  raised  by  floating  a  stick  L  into 
the  position  Zi ,  where  it  is  supported  against  the 

verticals.  The  stick  then  falls  of  itself  into  position 
Li  and  rests  on  the  crest.  Explain  the  reason 
of  this. 

40.  A  sluice  3  ft.  square  and  with  a  head  of  12 
ft.  over  the  centre  has,  from   the  thickness  of  the 
frame,  the  contraction  suppressed  on  all  sides  when 
fully  open ;   when   partially   open,  the    contraction 

exists  on  the  upper  edge,  i.e.,  against  the  bottom  of  the  gate,  which  is 
formed  of  a  thin  sheet  of  metal.  Find  the  discharge  in  cubic  feet  when 
opened  i  ft.,  2  ft.  and  also  when  fully  open.  Ans.  57.77  ;  114.45  ;  '75-9. 


66  HYDRA  ULICS. 

41.  What  quantity  of  water  flows  through  the  vertical  aperture  of  a 
dam,  its  width  being  36  in.  and  its  depth  10  in. ;  the  upper  edge  of  the 
aperture  is  16  ft.  below  the  surface.  Ans.  50.65  cub.  ft.  per  sec. 

42.  264  cubic  feet  of  water  are  discharged  through  an  orifice  of  5  sq. 
ins.  in  3  min.  10  sec.     Find  the  mean  velocity  of  efflux. 

Ans.  64  ft.  per  sec. 

43.  One  of  the  locks  on  the  Lachine  Canal  has  a  superficial  area  of 
about  12,150  sq.  ft.,  and  the  difference  of  level  between  the  surfaces  of 
the  water  in  the  lock  and  in  the  upper  reach  is  9  feet.     Each  leaf  of  the 
gates  is  supplied  with  one  sluice,  and  the  water  is  levelled  up  in  2  min, 
48  sees.     Determine  the  proper  area  of  the  sluice-opening.     (Centre  of 
sluice  20  ft.  below  surface  of  upper  reach.) 

Ans.  Area  of  one  sluice  =  43.73  sq.  ft. 

44.  The  horizontal  section  of  a  lock-chamber   may  be   assumed  a 
rectangle,  the  length  being  360  ft.     When  the  chamber  is  full,  the  sur- 
face width  between  the  side  walls,  which  have  each  a  batter  of  i  in  12, 
is  45  ft.     How  long  will  it  take  to  empty  the  lock  through  two  sluices  in 
the  gates,  each  8  ft.  by  2  ft.,  the  height  of  the  water  above  the  centre  of 
the  sluices  being  13  feet  in  the  lock  and  4  feet  in  the  canal  on  the  down- 
stream side.  Ans.  594  sec.,  c  being  .625. 

45.  Water  approaches  a  rectangular  opening  2  ft.  wide  with  a  velocity 
of  4  ft.  per  second.     At  the  opening  the  head  of  water  over  the  lower 
edge  =  13  ft.,  and  over  the  surface  of  the  tail-race  =  12  ft.;  the  discharge 
through  the  opening  is  70  cub.  ft.  per  second.     Find  the  height  of  the 
opening.  Ans.  1.022  ft. 

46.  The  water  in  a  regulating-chamber  is  8  ft.  below  the  level  of  the 
water  in  the  canal  and  8  ft.  above  the  centre  of  the  discharging-sluice. 
Determine  the  rise  in  the  canal  which  will  increase  the  discharge  by  10 
per  cent.  Ans.  1.68  ft. 

The  horizontal  sectional  area  of  the  chamber  is  constant  and  equal  to 
400  sq.  ft.;  in  what  time  will  the  water  in  the  chamber  rise  to  the  level  of 
that  in  the  canal,  if  the  discharging-sluice  is  closed;  the  sluice  between 
the  canal  and  chamber  being  3  sq.  ft.  in  area?  Ans.  150.83  sec. 

47.  A  lock  on  the  Lachine  Canal  is  270  ft.  long  by  45  ft.  wide  and  has 
a  lift  of  8£  ft.;  there  are  two  sluices  in  each  leaf,  each  8f  ft.  wide  by 
2£  ft.  deep ;  the  head  over  the  horizontal  centre  line  of  the  sluices  is 
19  ft.     Find  the  time  required  to  fill  the  lock.  Ans.  164.6  sec. 

48.  Show  that  the  energy  of  a  jet  issuing  through  a  large  rectangular 
orifice  of  breadth  B  is  i2$B(ff£  —  Hi*),  Hi ,  H*  being  the  depths  below 
the  water-surface  of  the  upper  and  lower  edges  of  the  orifice,  and  the 
coefficient  of  discharge  being  .625. 

49.  A  reservoir  at  full  water  has  a  depth  of  40  ft.  over  the  centre  of 
the  discharging-sluice,  which  is  rectangular  and   24  in.  wide  by  18  in. 
deep.    Find  the  discharge  in  cubic  feet  per  second  at  that  depth,  and  also 


FLOW    THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  6? 

when  the  water  has  fallen  to  30,  20,  and  10  ft.,  respectively;  find  the 
mechanical  effect  of  the  efflux  in  each  case. 

Ans.  94.8   cub.  ft.;  82.1  cub.  ft.;  67  cub.  ft.;  47.4  cub.  ft.;  431.2 
H.P.;  280  H.P.;  152.5  H.P.;  53.95  H.P. 

50.  Require  the  head  necessary  to  give  7.8  cubic  feet  per   second 
through  an  orifice  36  sq.  in.  in  sectional  area.  Aps.  38.9  ft. 

51.  The  upper  and  lower  edges  of  a  vertical  rectangular  orifice  are 
6  and  10  feet  below  the  surface  of  the  water  in  a  cistern,  respectively ; 
the  width  of  the  orifice  is  i  ft.     Find  the  discharge  through  it. 

Ans.  5642  cub.  ft.  per  sec. 

52.  To  find  the  quantity  of  water  conveyed  away  by  a  canal   3  ft. 
wide,  a  board  with  an  orifice  2  ft.  wide  and  i  ft.  deep  is  placed  across 
the  canal  and  dams  it  back  until  it  attains  a  height  of  2£  ft.  above  the 
bottom  and  if  ft.  above  the  lower  edge  of  the  orifice.     Find  the   dis- 
charge,    (c  =  .625.)  Ans.  17.59  cub-  ft-  per  sec. 

53.  Six  thousand  gallons  of  water  per  minute  are   forced  through  a 
line  of  piping  ABC  and  are  discharged  into  the  atmosphere  at  C,  which 
is  6  ft.  vertically  above  A.     The  pipe  AB  is  12  in.  in  diameter  and  12  ft. 
in  length  ;  the  pipe  J5C  is  6  in.  in  diameter  and  12  ft.  in  length.     Disre- 
garding friction,  find  the  "  loss  in  shock  "  and  draw  the  plane  of  charge. 

Ans.  Loss  of  head  in  shock  =  57.9  ft. 

54.  What  should  be  the  height  of  a  drowned   weir  400  ft.  long,  to 
deepen  the  water  on  the  up-stream  side  by  50  per  cent,  the  section  of 
the  stream  being  400  ft.  x  8  ft.,  and  the  velocity  of  approach  3  ft.  per 
second  ?  Ans.  8.396  ft. 

55.  The  two  sluices  each  4  ft.  wide  by  2  ft.  deep  in  a  lock-gate  are 
submerged  one  half  their  depth.     The  constant  head  of  water  above  the 
axis  of  the  sluice   is  12   ft.      Find   the  discharge  through  the  sluice, 
the  velocity  of  approach  being  4  ft.  per  second. 

Ans.  16626.2  cub.  ft.  per  minute. 

56.  Find  the  flow  through  a  square  opening,  one  diagonal  being  ver- 
tical and  12  in.  in  length,  and  the  upper  extremity  of  the  diagonal  be- 
ing in  the  surface  of  the  water.  Ans.  1.727  cub.  ft.  per  sec. 

57.  The  locks  on  the  Montgomeryshire  Canal  are  81  ft.  long  and  7f 
ft.  wide  ;  at  one  of  the  locks  the  lift  is  7  ft.;  a  24-in.  pipe  leads  the  water 
from  the  upper  level  and  discharges  below  the  surface  of  the  lower  level 
into  the  lock-chamber ;  the  mouth  of  the  pipe  is  square,  2  ft.  in  the  side, 
and  gradually  changes  into  a  circular  pipe  2  ft.  in  diameter.     Find  time 
of  filling  the  lock,     (c  =  i.)  Ans.  130  sees. 

58.  A  canal  lock  is  115.1  ft.  long  and   30.44  ft.   wide;  the  vertical 
depth  from  centre  of  sluice  to  lower  reach  is  1.0763  ft.,  the  charge  being 
6.3945  ft.  ;  the  area  of  the  two  sluices  is  2  x  6.766  sq.  ft.     Find  the  time 
of  filling  up  to  centre  of  sluices,     (c  =  .625  for  the  sluice,  but  is  reduced 


68  HYDRA  ULICS. 

to  .548  when  both  are  opened.)     Also,  find  time  of  filling  up  to  level  of 
upper  reach  from  centre  of  sluice-doors.  Ans.  25  sec.;  298  sees. 

59.  A  reservoir  half  an  acre  in  area  with  sides  nearly  vertical,  so  that 
it  may  be  considered  prismatic,  receives  a  stream  yielding  9  cub.  ft.  per 
second,  and  discharges  through  a  sluice  4  ft.  wide,  which  is  raised  2  ft. 
Calculate  the  time  required  to  lower  the  surface  5  ft.,  the  head  over  the 
centre  of  the  sluice  when  opened  being  10  feet.  Ans.  1079  sees. 

60.  Show  that  in  a  channel  of  V  section  an  increment  of  10  per  cent 
in  the  depth  will  produce  a  corresponding  increment  of  5  per  cent  in  the 
velocity  of  flow  and  of  25  per  cent  in  the  discharge. 

61.  The  angle  of  a  triangular  notch  is  90°.     How  high  must  the 
water  rise  in  the  notch  so  that  the  discharge  may  be  1000  gallons  per 
minute?  Ans.  1 2  inches  very  nearly. 

62.  Show  that  upon  a  weir  10  feet  long  with  12  inches  depth  of  water 
flowing  over,  an  error  of  i/iooo  of  a  foot  in  measuring  the  head  will 
cause  an  error  of  3  cubic  feet  per  minute  in  the  discharge,  and  an  error 
of  i/ioo  of  a  foot  in  measuring  the  length  of  the  weir  will  cause  an  error 
of  2  cubic  feet  in  the  discharge. 

63.  In  the  weir  at  Killaloe  the  total  length  is  noo  ft.,  of  which  779  ft. 
from  the  east  abutment  is  level,  while  the  remainder  slopes  i  in  214,  giving 
a  total  rise  at  the  west  abutment  of  1.5  ft.    Calculate  the  total  discharge 
over  the  weir  when  the  depth  of  water  on  the  level  part  is  1.8  ft.,  which 
gives  .3  ft.  on  highest  part  of  weir.     (Divide  slope  into  8  lengths  of  40 
ft.  each,  and  assume  them  severally  level,  with  a  head  equal  to  the 
arithmetic  mean  of  the  head  at  the  beginning  and  end  of  each  length.) 

Ans.  7483  cub.  ft.  per  sec. 

64.  A  watercourse  is  to  be  augmented  by  the  streams  and  springs 
above  its  level.     The  latter  are  severally  dammed  up  at  suitable  places 
and  a  narrow  board  is  provided  in  which  an  opening  12  in.  long  by  6 
in.  deep  is  cut  for  an  overfall ;  it  was  surmised  that  this  would  be  suf- 
ficient for  the  largest  streams;   another  piece  attached  to  the  former 
would  reduce  the  length  to  6  in.  for  smaller  streams.     Calculate  the 
delivery  by  the  following  streams: 

In  No.  i  stream  with  the  12-in.  notch,  depth  over  crest  =  .37  ft. 
"   No.  2      "          "       "      6-in.      "  "         "        "      =  .41  ft.       -; 

"   No.  3      "          "       "    12-in.      "  "         "        "      =  .29  ft. 

"    No.  4      "          "       "      6  in.      "  "         "        "      =  .19  ft. 

(Take  into  account  the  side  contractions.) 

Ans.  No.  i,  .695  cub.  ft. ;  No.  2,  .3658  cub.  ft. ;  No.  3,  .4904  cub. 
ft.;  No.  4,  .1275  cub.  ft. 

65.  The  horizontal  sectional  area  of  a  reservoir  is  constant  and  = 
10,000  square  feet.     When  the  reservoir  is  full,  a  right-angled  notch  2 
ft.  deep  is  opened.     Find  the  time  in  which  the  level  of  the  water  falls- 
to  the  bottom  of  the  notch.  Ans.  15.3  min. 


FLOW   THROUGH  ORIFICES,   OVER   WEIRS,  ETC.  69 

66.  A  weir  passes  6  cubic  feet  per  second,  and  the  head  over  the  crest 
is  8  inches.     Find  the  length  of  the  weir.  Ans.  3.3068  ft. 

67.  A  weir  400  ft.  long,  with  a  9-in.  depth  of  water  on  it,  discharges 
through  a  lower  weir  500  ft.  long.    Find  the  depth  of  water  on  the  latter. 

Ans.  .6457  ft. 

68.  A  stream  30  ft.  wide,  3  ft.  deep,  discharges  310  cubic  feet  per 
second  ;  a  weir  2  feet  deep  is  built  across  the  stream.     Find  increased 
depth  of  latter,  (a)  neglecting  velocity  of  approach,  (b)  taking  velocity 
of  approach  into  account.  Ans.  (a)  1.26  ft.  to  1.265  ft-J 

(6)  1. 19  ft. 

69.  A  weir  is  545  ft.  long;  how  high  will  the  water  rise  over  it  when 
it  rises  .68  ft.  upon  an  upper  weir  750  ft.  long?  Ans.  .8413  ft. 

70.  In  a  stream  50  ft.  wide  and  4  ft.  deep  water  flows  at  the  rate  of 
loo  ft.  per  minute ;  find  the  height  of  a  weir  which  will  increase  the 
depth  to  6  ft.,  (i)  neglecting  velocity  of  approach,  (2)  taking  velocity  of 
approach  into  account.  Ans.  (i)  4.4126  ft;  (2)  4.4509  ft. 

71.  A  stream  50  ft.  wide   and  4  ft.  deep  has  a  velocity  of  3  ft.  per 
second ;  find  the  height  of  the  weir  which  will  double  the  depth,  (i) 
neglecting  velocity  of  approach,  (2)  taking  velocity  of  approach  into  ac- 
count. Ans.  (i)  5.615  ft.;  (2)  5.7688  ft. 

72.  A  stream  80  ft.  wide  by  4  ft.  deep  discharges  across  a  vertical 
section  at  the  rate  of  640  cubic  feet  per  second  ;  a  weir  is  built  in  the 
stream,  increasing  its  depth  to  6  ft.     Find  the  height  of  the  weir. 

Ans.  4.233  ft. 

73.  Salmon-gaps  are  constructed  in  a  weir  ;  they  are  each  10  ft.  wide 
and  their  crests  are  18  in.  below  the  weir  crest.     Calculate  the  discharge 
down  three  of  these  gaps,  the  water  on  the  level  rjart  of  the  weir  being 
8  in.  deep.  Ans.  238.15  cub.  ft.  per  sec. 

74.  A  pond  whose  area  is  12,000  sq.  ft.  has  an  overfall  outlet  36  in. 
wide,  which  at  the  commencement  of  the  discharge  has  a  head  of  2.8  ft. 
Find  the  time  required  to  lower  the  surface  12  in.         Ans.  354.72  sec. 

75.  How  much  water  will  flow  in  an  hour  through  a  rectangular 
notch  24  in.  wide,  the  surface  of  still  water  being  8  in.  above  the  crest 
of  the  notch  ?     (Take  into  account  side  contraction.)       Ans.  3.386  ft. 

76.  Show  that   when  the  water  flowing  over  has  a 
depth   greater  than   .3874   ft.  it   is  carried  completely 
over   the   longitudinal    opening,  .83   ft.   in   width.     At 
what  depth  does  all  the  water  flow  in  ?  , 

Ans.  .221  ft.  FIG.  49. 


CHAPTER  II. 


FLUID  FRICTION. 

I.  Fluid  Friction. — The  term  fluid  friction  is  applied  to 
the  resistance  to  motion  which  is  developed  when  a  fluid  flows 
over  a  solid  surface,  and  is  due  to  the  viscosity  of  the  fluid. 
This  resistance  is  necessarily  accompanied  by  a  loss  of  energy 
caused  by  the  production  of  eddies  along  the  surface,  and 
similar  to  the  loss  which  occurs  at  an  abrupt  change  of  sec- 
tion, or  at  an  angle  in  a  pipe  or  channel. 

Froude's  experiments  on  the  resistance  to  the  edgewise 
motion  of  planks  in  a  fluid  mass,  the  planks  being  T\  in.  thick, 
19  in  deep,  and  I  to  50  ft.  long,  each  plank  having  a  fine  cut- 
water and  run,  are  summarized  in  the  following  table : 


Length  of  Surface  in  Feet. 

Nature  of  Surface 

2  Feet. 

8  Feet. 

20  Feet. 

50  Feet. 

Covering. 

A 

B 

c 

A 

B 

C 

A 

B 

C 

A 

B 

C 

Varnish  

2.OO 

•  41 

•  3QO 

1.85 

.325 

.264 

1.85 

.278 

.240 

I.83 

.250 

.226 

Paraffine  

38 

•  37O 

1.94 

.314 

?6o 

1.93 

.271 

.237 

Tinfoil 

2    16 

qo 

2QC 

I  »QQ 

.278 

.263 

I  .QO 

?6^ 

•244 

r  83 

.246 

272 

Calico                    .  .  . 

i  cn 

87 

•  72^ 

I  .Q2 

.626 

.504 

I.  80 

.  C.T.I 

•447 

T  87 

.474 

•  42^ 

Fine  sand  

2.00 

.81 

.690 

2.00 

.583 

•450 

2.00 

.480 

•  384 

2.06 

•405 

•  337 

Medium  sand  

2.OO 

.90 

.7302.00 

.625 

.488 

2.00 

•534 

•  465 

2.OO 

.488 

•456 

Coarse  sand  

2.00 

I.  10 

.880 

2.OO 

.714 

.520 

2.0O 

.588 

.490 

Columns  A  give  the  power  of  the  speed  (v)  to  which  the  re- 
sistance is  approximately  proportional. 

Columns  B  give  the  mean  resistance  in  Ibs.  per  square  foot  of 
the  whole  surface  of  a  board  of  the  lengths  stated  in  the  table. 

Columns  C  give  the  resistance,  in  pounds,  of  a  square  foot 
of  surface  at  the  distance  sternward  from  the  cutwater  stated  in 

70 


FLUID   FRICTION.  7 1 

the  heading,  each  plank  having  a  standard  speed  of  10  ft.  per 
second.  The  resistance  at  other  speeds  can  be  easily  calculated. 

An  examination  of  the  table  shows  that  the  mean  resistance 
per  square  foot  diminishes  as  the  length  of  the  plank  increases. 
This  may  be  explained  by  the  supposition  that  the  friction  in 
the  forward  portion  of  the  plank  develops  a  force  which  drags 
the  water  along  with  the  surface,  so  that  the  relative  velocity 
of  flow  over  the  rear  portion  is  diminished.  Again,  the  de- 
crease of  the  mean  resistance  per  square  foot  is  .132  Ib.  when 
the  length  of  a  varnished  plank  is  increased  from  2  to  20  ft.,  while 
it  is  only  .028  Ib.  when  the  length  increases  from  20  to  50  ft. 
Hence,  for  greater  lengths  than  50  ft.  the  decrease  of  resistance 
may  be  disregarded  without  much,  if  any,  practical  effect. 

Thus,  generally  speaking,  these  experiments  indicate  tha-t 
the  mean  resistance  is  proportional  to  the  #th  power  of  the 
relative  velocity,  n  varying  from  1.83  to  2.16,  and  its  average 
value  being  very  nearly  2. 

Colonel  Beaufoy,  as  a  result  of  experiments  at  Deptford, 
also  assumed  the  mean  resistance  to  be  proportional  to  the  nth 
power  of  the  relative  velocity,  the  value  of  n  in  three  series  of 
observations  being  1.66,  1.71  and  1.9. 

The  frictional  resistance  is  evidently  proportional  to  some 
function  of  the  velocity,  F(v),  which  should  vanish  when  v  is 
nil,  as  when  the  surface  is  level,  and  should  increase  with  v. 

Coulomb  assumed  the  function  F(v)  to  be  of  the  form 
av  -f-  bv* ,  a  and  b  being  coefficients  to  be  determined  by  experi- 
ment. Experiment  shows  that  when  v  does  not  exceed  5  ft. 
per  minute  the  resistance  is  directly  proportional  to  the  veloc- 
ity, but  that  it  is  more  nearly  proportional  to  the  square  of  the 
velocity  when  the  velocity  exceeds  30  ft.  per  minute  ;  or, 

F(v)  =  av  when  v  <    5  ft.  per  minute, 
and 

F(v)  =  bv^  when  v  >  30  ft.  per  minute. 

Again,  observations  on  the  flow  of  water  in  town  mains 
indicate  that  no  difference  of  resistance  is  developed  under 


72  HYDRA  ULICS. 

widely  varying  pressures,  and  this  independence  of  pressure  is 
also  verified  by  Coulomb's  experiment  showing  that  if  a  disk 
is  oscillated  in  water  there  is  no  apparent  change  in  the  rate  of 
decrease  of  the  oscillations,  whether  the  water  is  under  atmos- 
pheric pressure  or  not. 

From  the  preceding  and  other  similar  experiments  the  fol- 
lowing general  laws  of  fluid  friction  have  been  formulated : 

(1)  The  frictional  resistance  is  independent  of  the  pressure 
between  the  fluid  and  the  surface  over  which  it  flows. 

(2)  The  frictional  resistance  is  proportional  to  the  area  of 
the  surface. 

(3)  The  frictional  resistance  is  proportional  to  some  func- 
tion, usually  the  square,  of  the  velocity. 

To  these  three  laws  may  probably  be  added  a  fourth,  viz.: 

(4)  The  frictional  resistance  is  proportional  to  the  density 
of  the  fluid. 

A  fifth  law,  viz.,  that  "  the  frictional  resistance  is  indepen- 
dent of  the  nature  of  the  surface  against  which  the  fluid  flows," 
has  been  sometimes  enunciated,  and  at  very  low  velocities 
the  law  is  approximately  true.  At  high  velocities,  however, 
such  as  are  common  in  engineering  practice,  the  resistance  has 
been  shown  by  experiment,  and  especially  by  the  experiments 
carried  out  by  Darcy,  to  be  very  largely  influenced  by  the 
nature  of  the  surface. 

Let  p  be  the  frictional  resistance  in  pounds  per  square  foot 
of  surface  at  a  velocity  of  I  ft.  per  second. 

Let  A  be  the  area  of  the  surface  in  square  feet. 

Let  v  be  the  relative  velocity  of  the  surface  and  the  water 
in  which  it  is  immersed. 

Let  R  be  the  total  frictional  resistance. 

Then  from  the  laws  of  fluid  friction 

R  =  p .  AV*. 

2j? 

Take/  =  —  p,  w  being  the  specific  weight  of  the  fluid.  Then 
R  =  fwA—. 


FLUID   FRICTION.  73 

The  coefficient  f  is  approximately  constant  for  any  given 
surface,  and  is  termed  the  coefficient  of  fluid  friction.  The 
power  absorbed  by  the  frictional  resistance 

v* 
=  pAv'  X  v  =  pAv*  =  fwA  — . 

o 


TABLE   GIVING  THE  AVERAGE  VALUES  OF  /  IN  THE   CASE  OF 

LARGE   SURFACES    MOVING   IN   AN  INDEFINITELY    LARGE 
MASS   OF   WATER. 

Surface.  Coefficient  of  Friction  (/"). 

New  well-painted  iron  plate  ..............  00489 

Painted  and  planed  plank   ...............  0035 

Surface  of  iron  ships   ...  .................  00362 

Varnished  surface  ........................  00258 

Fine  sand  surface  ........................  00418 

Coarse  sand  surface  ....  .................  00503 

2.  Surface  Friction  of  Pipes.  —  Assuming  that  the  laws  of 
fluid  friction  already  enunciated  hold  good  when  water  flows 
through  a  pipe,  it  has  been  shown  by  numerous  experiments 
that  the  coefficient  of  friction  /lies  between  the  limits  .005  and 
.01,  its  average  value  under  ordinary  conditions  being  about 
.0075.  No  single  value  of  f  is  applicable  to  very  different 
cases.  Indeed,  /depends  not  only  upon  the  condition  of  the 
surface,  but  also  upon  the  diameter  of  the  pipe  and  the  veloc- 
ity of  the  water.  Some  authorities  have  expressed  its  value  by 
a  relation  of  the  form 


a  and  b  being  constants  whose  values  are  to  be  determined  by 
experiment. 

The  following  table  gives  some  of  the  best  numerical  results 
obtained  for  a  and  b\ 


74  HYDRA  ULICS. 

Authority.                                a  b 

Prony   ........  ..........  00021230  .00003466 

D'Aubuisson  ............  0002090  .000037608 

Eytelwein  ...............  00017059  .00004441 

In  pipes  of  small  diameter  in  which  the  velocity  of  flow  is 

less  than  4  in.  per  second  the  term  a  may  be  disregarded  so 
that 


In  ordinary  practice  and  when  the  pipes  have  been  in  use 
for  some  time,  the  velocity  usually  exceeds  4  in.  per  second, 

and  the  term  —  may  then  be  disregarded,  so  that 


Now  Darcy's  experiments  have  shown  that  it  is  more  cor- 
rect to  assume  that  a  and  b,  instead  of  being  constant,  are 
variable,  and  Darcy  expressed  them  as  functions  of  the  diam- 
eter of  the  pipe. 

Thus,  for  pipes  in  which  the  velocity  exceeds  4  in.  per 
second,  Darcy  took 

/  ,  £ 

g   ''  ^d' 

d  being  the  diameter  of  the  pipe,  and  a  and  ft  coefficients. 
Darcy  also  gave  the  following  values  for  a  and  ft : 

a      '  ft 

For   drawn   wrought-iron   or   smooth 

cast-iron  pipes 0001545         .000012973 

For  pipes  with   surfaces    covered   by 

light  incrustations 0003093         .00002598 


FLUID   FRICTION. 


75 


These  coefficients  can  be  put  into  the  following  very  simple 
form  without  sensibly  altering  their  values  : 


For  clean  pipes  ...........     f=  .005(1  -[-  -  ) 

For  slightly  incrusted  pipes     /  =  .01(1  -f-  -  J 

d  being  the  diameter  in  feet. 

Darcy  proposed  to  include  all  cases  by  expressing  /"more 
generally  in  the  form 


in  which,  for  new  and  smooth  iron  pipes, 

a  =  ,00003959,  ft  —  .00002603125  ; 

af  =  .000064375,          ft'  =  .000000335625. 

These  values  are  rarely  of  any  practical  use. 


TABLE   GIVING   DARCY'S   VALUES   OF  /  FOR  VELOCITIES 
EXCEEDING  4  IN.  PER   SECOND. 


Diam. 

Value  of/ 

Diana. 

Value  of  f. 

Diam. 

Value  of/. 

of 

of 

of 

Pipe 

Pipe 

Pipe 

m 

New 

Incrusted 

in 

New 

Incrusted 

in 

New 

Incrusted 

Inches. 

Pipes. 

Pipes. 

Inches. 

Pipes. 

Pipes. 

Inches. 

Pipes. 

Pipes. 

2 

.0075 

.0150 

9 

.00556 

.OITII 

27 

.00519 

.01037 

3 

.00667 

•01333 

12 

.00542 

.01083 

30 

.00517 

.01033 

4 

.00625 

.0125 

15 

•00533 

.01067 

36 

.00514 

.01028 

5 

.0060 

.OI2 

18 

.00528 

.01056 

42 

.00512 

.OIO24 

6 

.00583 

.01167 

2.1 

.00524 

.01048 

48 

.00510 

.01021 

7 

.00571 

.01143 

24 

.00521 

.01042 

54 

.00509 

.OIOI9 

8 

.00563 

.01125 

76  HYDRAULICS. 

Again,  Weisbach  has  proposed  the  formula 


-* 

Vv 

where  a  =  .003598  and  b  =  .004289. 

3.  Resistance  of  Ships.  —  The  motion  of  a  ship  through 
water  causes  the  production  of  waves  and  eddies,  and  the  total 
resistance  to  the  movement  of  a  ship  is  made  up  of  a  frictional 
resistance,  a  wave-making  resistance,  and  an  eddy-making  re- 
sistance. Although  there  is  no  theory  by  which  the  resistance 
at  a  given  speed  of  a  ship  of  definite  design  can  be  absolutely 
determined,  Froude's  experiments  render  it  possible  to  make 
certain  inferences  and  furnish  some  useful  data. 

According  to  Froude,  the  frictional  resistance  is  sensibly 
the  same  as  that  of  a  rectangular  surface  moving  with  the  same 
speed,  of  the  same  length  as  the  ship  in  the  direction  of  motion, 
and  of  an  area  equal  to  the  immersed  surface  of  the  ship. 
Experiments  seem  to  indicate  that  as  the  speed  increases,  the 
frictional  resistance  of  well-designed  ships  with  clean  bottoms 
is  from  90  to  60  per  cent  of  the  total  resistance,  and  that  the 
percentage  is  greater  when  the  bottoms  become  foul. 

The  wave-making  resistance  is  especially  affected  by  the 
form  and  proportions  of  the  ship,  depending,  for  a  given 
length,  upon  the  proportions  of  the  entrance,  middle  body,  and 
run.  For  every  ship  there  is  a  limit  of  speed  below  which  the 
resistance  is  approximately  proportional  to  the  square,  of  the 
speed,  being  chiefly  due  to  friction,  and  beyond  which  it  in- 
creases more  rapidly  than  as  the  square. 

The  eddy-resistance  in  the  case  of  well-formed  ships  should 
not  exceed  about  10  per  cent  of  the  total  resistance,  and  is 
often  much  less. 

Froude's  law  of  resistance  may  be  enunciated  as  follows  : 

Let  /„  /,  be  the  lengths  of  a  ship  and  its  model. 

Let  Alt  A^  be  the  displacements  of  a  ship  and  its  model. 

Let  /?„  R^  be  the  resistances  of  a  ship  and  its  model  at  the 
.speeds  ivl  and  vt. 


FLUID   FRICTION.  7? 

Then,  if 

_i _i_  __     \  _ 

V    "  /£   ~~    /f  *' 
2         *s  ^a 

the  resistances  are  in  the  ratio  of 


Hence,  too,  the  H.  P.,  and  therefore  also  the  coal  consumption 
per  hour,  is  proportional  to  Rv,  that  is,  to 

A1      or      /5, 

and  the  coal  consumption  per  mile  is  proportional  to 

A     or  to     /3. 

Again,  R  is  proportional  to  /3  ; 
that  is,  to  /  X  /3  1 
that  is,  to  v*  X  ^  ; 

and  it  is  sometimes  convenient  to  express  the  resistance  irt 
pounds  in  the  form 


v  being  the  speed  in  knots,  A  the  displacement  in  tons,  and  k 
a  coefficient  depending  upon  the  type  of  ship  and  varying  from 
.55  to  .85  when  the  bottom  is  clean. 


CHAPTER    III. 
FLOW  OF  WATER  IN  PIPES. 

1.  Assumptions. — In  the  ordinary  theory  of  the  flow  of 
water  in  a  pipe  it  is  assumed  that  the  water  consists  of  thin 
plane  layers  perpendicular  to  the  axis  of  the  pipe,  that  each 
layer  is  driven  through  the  pipe  by  the  action  of  gravity  and  by 
the  difference  of  pressure  on  its  plane  faces,  and  that  the  liquid 
molecules  in  any  layer  at  any  given  moment  will  also  be  found 
in  a  plane  layer  after  any  interval  of  time.    In  such  motion  the 
internal  work  done  in  deforming  a  layer  may  be  generally  dis- 
regarded. 

It  is  further  assumed  that  there  is  no  variation  of  velocity 
over  the  surface  of  a  layer,  and  this  is  equivalent  to  saying  that 
each  liquid  molecule  in  a  cross-section  has  the  same  mean  ve- 
locity. 

The  disagreement  of  these  assumptions  with  the  results  of 
recent  experimental  researches  will  be  referred  to  in  a  subse- 
quent article. 

2.  Steady  Motion  in  a  Pipe  of  Uniform  Section. — Since 
the  motion  is  to  be  steady,  the  same  volume  Q  cub.  ft.  of  water 
will  always  arrive  at  any  given  cross-section  of  A  square  feet 
with  the  same  mean  velocity  v  ft.  per  second.     Then 

Q  =  Av. 

But  since  the  pipe  is  of  constant  diameter,  A  is  constant,  and 
hence  also  v  is  constant,  so  that  the  mean  velocity  is  the  same 
throughout  the  whole  length  of  the  pipe. 

Consider  an  elementary  mass  of  the  fluid  AABB,  bounded 
by  the  pipe  and  by  the  two  cross-sections  AA,  BB.  Let  dl 

78 


FLOW  OF    WATER  IN  PIPES. 


79 


FIG.  50. 


be  the  length  AB  of  the   element,  the   length  /  ft.  of   the 
pipe  being  measured  along  the 
axis  from  any  origin  O. 

Let  z,  z  +  dz  be  the  eleva- 
tions in  feet  above  a  datum 
line  of  the  centres  of  pressure 
in  the  cross-sections  A  A,  BB, 
respectively. 

Let  p,  p  +  dp  be  the  intens- 
ities of  the  pressures  on  these 
cross-sections   in   pounds    per 
square  foot. 

Let  P  be  the  perimeter  of 
the  pipe. 

Let  w  be  the  specific  weight  of  the  water  in  pounds  per 
cubic  foot. 

Work  Done  by  Gravity. — In  one  second  wQ  Ibs.  of  water 
are  transferred  from  AA  to  BB,  falling  through  a  vertical  dis- 
tance of  dz  ft.  Thus  the  work  done  by  gravity  per  second 

=  —  wQ  .  dz, 

a  positive  quantity  if  dz  is  negative,  and  vice  versa. 

Work  Done  by  Pressure. — The  total  pressure  on  A  A  paral- 
lel to  the  axis  =  pA  ;  the  total  pressure  on  BB  parallel  to 
the  axis  =  (p  +  dp) A. 

Therefore  ^the  total  resultant  pressure  parallel  to  the  axis 
in  the  direction  of  motion  =  —  A  .  dp,  and  the  work  done  per 
second  on  the  volume  Q  by  this  pressure  =  —  Q  .  dp. 

Note. — The  work  done  by  the  pressure  at  the  pipe  surface  is  nil,  as 
its  direction  is  at  right  angles  to  the  line  of  motion. 

Work  Absorbed  by  Frictional  Resistance. — From  the  laws  of 
fluid  friction  this  work  per  second  is  evidently 

p 

—  —  P .  dl .  F(v)  X  v  = -r  .  Q  .  F(v)  .  dl, 


the  sign  being  negative  as  the  work  is  done  against  a  resistance. 


80  HYDRA  ULICS. 

Since  the  motion  is  steady,  the  work  done  by  the  external 
forces  must  be  equivalent  to  the  work  absorbed  by  the  fric- 
tional  resistance,  and  hence 

—  wQ  .  dz  —  Q  .  dp  -    —Q  .  F(v)  .  dl  —  o, 

or 

,   dp         P    F(v}     „ 

(Jn         I  *  _l          V       '  /y/     Q 

w         A  "    w 
Integrating, 


^  +        -f-j../=a  constant  =  H, 

w         A       w 

so  that  H  ft.-lbs.  per  pound  of  fluid  is  the  uniformly  distributed 
total  constant  energy. 

A 

—  is  called  the  hydraulic  mean  radius  of  a  pipe  and  will  be 

denoted  by  m. 
Take 


W  2g 

the  value  adopted  in  ordinary  practice,  f  being  the  coefficient 
of  friction.     Then 


w         in  2g 

Let  #,  ,  Al  ,/,  be  the  elevation  above  datum,  the  area  of  the 
cross-section,  and  the  intensity  of  the  pressure 
at  any  point  X  on  the  axis  of  the  pipe  distant 
/x  from  the  origin  (Fig.  51). 

Let  £2,  AS,  pi  be  the  elevation  above  datum,  the  area  of  the 
cross-section,  and  the  intensity  of  the  pressure 
at  any  other  point  Y  on  the  axis  distant  4 
from  the  origin  (Fig.  51). 


FLO W  OF   WATER  IN  PIPES. 
Then,  from  the  equation  just  deduced, 


81 


,.+*  +  £*=*=*+>•  +  £«!. 

w        m   2g  w       m  2g 


Hence 


w 


m  2g 


L  being  the  length  /2  —  ^  of  the  pipe  between  the  two  points 
K 


FIG.  51. 

Let  vertical  tubes  (pressure-columns)  be  inserted  in  the«.pipe 
at  X  and  at  Y.  The  water  will  rise  in  these  tubes  to  the  levels 
C  and  Z>,  and  evidently 


°  being  the  intensity  of  the  atmospheric  pressure. 


82  HYDRA  ULICS. 

Hence,  if  CX  and  D Fare  produced  meet  the  datum  line  in 
E  and  F, 

I  A  i/^j^iA       ri?  j  A 

#.  -+-  —  =  -Sj  -h  CA  -f-  —  =  Czi  -t-  -JL 

ze;  w  w 

and 

#a  +  —  =  £a  +  jC>F+  —  =  Z>F+— . 
w  w  w 

Therefore 


w  wi  m  2g 

G  being  the  point  in  which  the  horizontal  through  C  meets  FD 
produced. 

DG  is  called  the  "  virtual  fall  "  of  the  pipe,  being  the  fall  of 
level  in  the  pressure-columns;  and  since  there  would  be  no  fall. 
of  level  if  the  friction  were  nil,  DG  is  said  te  be  the  head  lost 
in  friction  in  the  distance  XY. 

Denote  this  head  by  h\  then 


= 

m  2g 

and  therefore 

£_/»• 

L       m  2g 

This  ratio  -  is  designated  the  virtual  slope  of  the  pipe,  and 

JL/ 

is   the  head  lost  in   friction  per  unit  of   length      It  will  be 
denoted  by  *',  so.  that 


If  the  section  of  the  pipe  is  a  circle  of  diameter  d,  or  a 
square  with  a  side  of  length  d,  then 


and 


FLOW  OF   WA7^ER   IN  PIPES.  t   _    83 

A         d 


__      = 

L  ~  d  2g 


3.  Influence  upon  the  Flow  of  the  Pipe's  Position  and 
Inclination.  —  In  Fig.  5  1  join  CD.  Now  since  the  fall  of  level 
(h)  is  proportional  to  Ly  the  free  surface  in  any  other  column 
between  X  and  Y  must  also  be  on  the  line  CD.  Thus  the 
pressure/7  at  any  intermediate  point  M  distant  x(==.  XM)  from 
X  is  given  by 


w  w  w 

Hence,  at  every  point  of  a  pipe  laid  below  CD,  the  fluid  pres- 
sure (pr)  exceeds  the  atmospheric  pressure  (/0)  by  an  amount 
w  .  MN,  so  that  if  holes  are  made  in  such  a  pipe  the  water  will 
flow  out  and  there  will  be  no  tendency  on  the  part  of  the  air 
to  flow  in.  In  pipes  so  placed  vertical  bends  may  be  intro- 
duced, care  being  taken  to  provide  for  the  removal  of  the  air 
which  may  collect  in  the  upper  parts  of  the  bends. 

If  the  line  of  the  pipe  coincides  with  CD,  i.e.,  with  the  vir- 
tual slope  or  line  of  free  surface  level,  MN  =  o,  and  the  fluid 
pressure  is  equal  to  that  of  the  atmosphere.  If  holes  are  now 
made  in  the  pipe  it  can  easily  be  shown  by  experiment  that 
there  will  be  neither  any  tendency  on  the  part  of  the  water  to 
flow  out  nor  on  the  part  of  the  air  to  flow  in. 


Next  take  CC'  =  DD'  =        and  join  CD'. 

w  J 


If  the  pipe  is  placed  in  any  position  between  CD  and  C ' Dr 
MN  becomes  negative,  and  the  fluid  pressure  in  the  pipe  is  less 
than  that  of  the  atmosphere.  If  holes  are  made  in  this  pipe, 
there  will  be  no  tendency  on  the  part  of  the  water  to  flow  out, 


84  HYDRA  ULICS. 

but  the  air  will  flow  in.  Thus,  if  a  pipe  rises  above  the  line  of 
virtual  slope,  there  is  a  danger  of  air  accumulating  in  the  pipe 
and  impeding,  or  perhaps  wholly  stopping,  the  flow.  No  verti- 
cal bends  should  be  introduced,  as  the  air  is  easily  set  free  and 
would  collect  in  the  upper  parts  of  the  bends,  with  the  effect 
of  impeding  the  flow  and  of  acting  detrimentally  upon  the  water 
itself,  which  the  liberation  of  the  air  renders  less  wholesome. 
If  the  line  of  pipe  coincide  with  CD',  then  the  fluid  pressure 
is  nil. 

Finally,  if  the  pipe  at  any  point  rises  above  CD',  the  press- 
ure becomes  negative,  which  is  impossible.  In  fact,  the  con- 
tinuity of  flow  is  destroyed,  and  the  pipe  will  no  longer  run  full 
bore.  Air  will  be  disengaged  and  will  rise  and  collect  at  the 
point  in  question,  so  that  in  order  to  prevent  the  flow  being 
wholly  impeded,  it  will  be  necessary  to  introduce  an  air-chamber 
at  this  point  from  which  the  air  can  be  removed  when  required. 


Note. — In  the  preceding  it  has  been  assumed  that  the  pipe  is  straight. 
If  the  pipe  is  curved,  so  also  is  the  line  of  virtual  slope.  In  ordinary 
practice,  however,  the  vertical  changes  of  level  in  a  pipe  at  different 
points  are  small  as  compared  with  the  length  of  the  pipe,  and  distances 
measured  along  the  pipe  are  sensibly  proportional  to  distances  measured 
along  the  horizontal  projection  of  the  pipe.  Hence  the  line  of  virtual 
slope  may  be  assumed  to  be  a  straight  line  without  error  of  practical 
importance. 


4.  Transmission  of  Energy  by  Hydraulic  Pressure.  — 
Let  Q  cub.  ft.  of  water  per  second  be  driven  through  a 
pipe  of  diameter  d  ft.  and  length  L  ft.  under  a  total  head  of 
H  ft.  Also  let  n  per  cent,  of  the  total  head  be  absorbed  in 
overcoming  the  frictional  resistance  in  the  pipe.  Then 

the  head  expended  in  useful  work  =  H  —  h 


H-h 
and  the  efficiency  = 


FLOW  OF   WATER  IN  PIPES.  8$ 

Again, 


—  -   h  -          -  - 

100  :      ~d~  2g  ~'' 


Since  Q  =  —  z/,  and  g  is  assumed  to  be  32,  thus, 


^     InHd* 

-  toy  T£~' 

ancf  the  work  transmitted  in  foot-pounds  per  second 


14 

If  ^V=  the  number  of  horse-power  transmitted,  then 


jv  -  _L  i;_5   /^£     1    A^8^ 
"550  H  V  /^    "28V  "7^~' 

and  this  equation  also  gives  the  distance  L  to  which  TV  horse- 
power can  be  transmitted  with  a  loss  of  n  per  cent  of  the  total 
head. 

Again, 

ffi  .  //  2fL  V*  2/Lw  v* 

the  efficiency  =  I  —  —  =  I :  -77  T  —  l  —  -^— 

H  gH  d  g     pd> 

p(  =  wH)  being  the  pressure  corresponding  to   the  head  H. 

Thus,  the  efficiency  is  constant  if  — -  is  constant. 

pa 

Assuming  this  to  be  the  case,  take  v*  =  c*  .pd.     Then  the 

total  energy  transmitted  =  wQH '  =  w vH 

4 


86  HYDRA  ULICS. 

If  it  be  also  assumed  that  the  thickness  /  of  the  pipe-metal 
is  so  small  that  the  formula 

pd  =  2ft 

holds  true,  f  being  the  circumferential  stress  induced  in  the 
metal,  then 

the  energy  transmitted  =  — 


F  being  the  volume  of  the  pipe  per  unit  of  length. 

Hence,  for  a  given  volume  (V)  of  metal  and  a  constant 
efficiency,  the  energy  transmitted  is  a  maximum  when  pd  is  a 
maximum. 

If  /  is  increased  beyond  a  certain  limit,  the  ratio  -^  is  no 

longer  small  and  the  thickness  t  will  have  a  greater  value 
than  that  given  by  the  equation  pd  =  2.  ft.  Then  the  cost  of 
the  pipe  will  also  increase.  On  the  other  hand,  if  d  is  increased 

the  ratio  -^,  and  therefore  also  the  pressure/,  will  remain  small, 

and  thus  the  cost  of  the  pipe  will  not  increase.  Hence  it  is 
more  economical  to  employ  large  pipes  and  low  pressures  than 
small  pipes  and  high  pressures. 

Note.  —  The  efficiency  diminishes  as  v  increases,  so  that,  as  far  as  the 
efficiency  is  concerned,  it  is  advantageous  to  transmit  the  energy  at  a 
low  speed. 

5.  Flow  in  a  Pipe  of  Uniform  Section  and  of  Length  Z, 
connecting  two  Reservoirs  at  Different  Levels.  —  Let  z  ft. 

be  the  difference  of  level  between  the  water-surface  in  the  two 
reservoirs. 


FLO W  OF   WATER   IN  PIPES.  8? 


FIG.  52. 

The  work  done  per  second  is  evidently  equal  to  the  work 
done  by  the  fall  of  wQ  pounds  of  water  through  the  vertical 
distance  z,  and  is  expended— 

(1)  In  producing  the  velocity  of  flow  v  feet   per   second 

which  requires  a  head  of  zl  feet  and  an  expenditure 
of  wQzl  foot-pounds  of  work  per  second  ; 

(2)  In  overcoming  the  resistance  at  the  entrance  from  the 

upper  reservoir  into  the  pipe,  which  requires  a  head 
of  sa  feet  and  an  expenditure  of  wQz^  foot-pounds 
of  work  per  second. 

(3)  In  overcoming  the  frictional  resistance  which  requires  a 

head  of  z^  feet  and  an  expenditure  of  wQz^  foot- 
pounds of  work  per  second.  Thus 


wQz  =  wQzl  +  wQz^  + 
or 

z  =  z        *        *. 


Now  #,  —  -  -  feet,  and  the  corresponding  energy  wQz^  is 

ultimately  wasted  in  producing  eddy  motions,  etc.,  in  the 
lower  reservoir. 

v* 
z^  may  be  expressed  in  the  form  n  —  feet,  n  being  a  coeffi- 

cient whose  value  varies  with  the  nature  of  the  construction  of 
the  entrance  into  the  pipe.  If  the  pipe-entrance  is  bell-mouth 
in  form,  n  =  .08,  but  if  it  is  cylindrical,  n  =  .5.  Finally, 


88  HYDRA  ULICS. 


f,  =          ft., 


, 

m     w  d    2g 

F(v)         v* 
Baking  -  -  =f  —  ,  as  is  usual  in  practice.     Hence 


2g\  d 


since  Q  =  --  v,  and  g  is  assumed  to  be  32. 

4 

For  given  values  of  Q  and  z  a  first  approximate  value  of  d 
may  be  obtained  from  the  last  equation  by  neglecting  the  term 

Q* 

—  rr;(l  +  «)•     Call  this  value  dv  and  substitute  it  for  the  d  in 

A/L 

the  term  —  j-  within  the  brackets.     A  second    approximation 
may  now  be  made  by  deducing  d  from  the  formula 


and  the  operation  may  be  again  repeated  if  desired. 

Generally  speaking,  I  +  n  is  usually  very  small  as  compared 


with       —  ,  and  may  be  disregarded  without  error  of  practical 

importance. 

The  formula  then  becomes 


_ 


which  is  known  as  Chezy's  formula  for  long  pipes. 

In  fact,  the  term  I  +  n  need  only  be  taken  into  account  in 
the  case  of  short  pipes  and  high  velocities. 


FLOW  OF   WATER   IN  PIPES. 


89 


6.  Losses  of  Head  due  to  Abrupt  Changes  of  Section, 
Elbows,  Valves,  etc. — When  the  velocity  or  the  direction  of 
motion  of  a  mass  of  water  flowing  through  a  pipe  is  abruptly 
changed,  the  water  is  broken  up  into  eddies  or  irregular  mo- 
tions which  are  soon  destroyed  by  viscosity,  the  corresponding 
energy  being  wasted. 

CASE  I.  Loss  due  to  a  sudden  contraction.     (Art.  16,  Chap.  I.) 
(a)  Let  water  flow  from  a  pipe  (Fig.  53),  or  from  a  reser- 
voir (Fig.  54)  into  a  pipe  of  sectional  area  A. 


FIG.  53- 


FIG.  54. 


Let  cc  be  the  coefficient  of  contraction. 

Then  the  area  of  the  contracted  section  =  ccA,  and 


the  loss  of  head  =  —  (--.  «,Y 
2     V         I 


2g  V, 


where  m  =  ( 2  I . 


=  m 


V2 
2? 


The  value  of  m  has  not  been  determined  with  any  great 
degree  of  accuracy ;  but  if  cc  =  .64,  then  m  =  .316. 


HYDRAULICS. 

When  the  water  enters  a  cylindrical  (not  bell-mouthed)  pipe 
from  a  large  reservoir,  the  value  of  m  is 
about  .505. 

(b)  Let  the  water  flow  across  the  abrupt 
change  of  section  through  a  central  ori- 
fice in  a  diaphragm  placed  as  in  Fig.  55. 
Let  a  be  the  area  of  the  orifice. 
Then  c,a  is  the  area  of  the  contracted  section,  and 


the  loss  of  head  =  ( — 
W 

(A      y 

where  m  =  I I J  • 

V^z         t 

According  to  Weisbach, 


I A          Vv*  v* 

=  I  —  —  I J  —  =  m  — , 
\cea         I  2g          2g' 


f-  = 

•I 

.2 

•3 

•4 

•5 

cc  = 

.616 

.614 

.612 

.610 

.607 

m  = 

231.7 

50.99 

19.78 

9.612 

5.256 

•i- 

.6 

•7 

.8 

•9 

I.OO 

Cc  — 

.605 

.603 

.601 

.598 

.596 

m  = 

3.077 

1.876 

1.169 

•734 

.48 

central 

~                                       =i    orifice   of   area  a.  nlaced   in 

a  cylin- 

-.V                 I 

drical  pipe  of 

7      1  

sectional  area 

A  as  in 

FIG.  56. 

Fig.  56. 

The  "  contracted  area  "  of  the  water  =  c<a  and 


the  loss  of  head 


i  IvA         V      v*  IA         \ 
=  —  I  ---  v)  =  --   --  i) 
2sr\cja         I       2.g\cfa        i 


=  m-—, 


where  m  =  { 


FLOW  OF   WATER  IN  PIPES.  9  1 

Generally  m  must  be  determined  by  experiment,  but  Weis- 
bach  gives  the  following  results  : 


if      =         -1                   -2                   .3                   .4  .5 

ce—       .624               .632               .643               .659  .681 

m=     225.9             4777            30.83            7.801  3.753 

if  ^  —         .6                  .7                  .8                  .9  i.  oo 

cc=       -712              .755               .813               .892  i.  oo 

m  =      1.796             .797               .29                .06  oo 

CASE  II.     Loss  due  to  a  Sudden  Enlargement.     (Fig.  57.) 

Let  Al  =  external  area  of  small  pipe. 

"    A,  =        "           "     "  large     " 
FIG.  57- 

r  ,    •  ,         i  fvA9          V        v*  (A,  V 

Then,  loss  of  head  =  —  \—-±  —  v\   =  —  -^-  —  i] 

*f\A.               I            2jr\A.  I 


2£- 

=  m — , 


(A          Y 
where  m  —  \-~  —  i)  . 


A 

Note.— The  losses  of  head  in  Case  I  (a)  and  in  Case  II  may  be 
avoided  by  substituting  a  gradual  and  regular  change  of  section  for  the 
abrupt  changes. 

CASE  III.  Loss  of  Head  due  to  Elbows.  (Fig.  58.)— The 
loss  of  head  due  to  the  disturbance  caused  by  an  elbow  is  ex- 

v* 
pressed  by  Weisbach  in  the  form  m — , 

o 

where  m  =.  9457  sin2  — +  2.047  sin4  —  > 

0  being  the  elbow  angle. 

Weisbach  deduced  this  formula  from  the  results  of  experi- 
ments with  pipes  1.2  in.  in  diameter. 


92 


HYDRA  ULICS. 


The  velocity  vl  with  which  the  water  flows  along  the  length 
AB  may  be  resolved  into  a  component  v  with  which  the  water 
flows  along  BC  and  a  component  u  at  right  angles  to  the 


FIG.  58. 

direction  of  v.      The  component  u  and   therefore   the   corre- 
sponding head,  viz., — ,  is  wasted.    The  component  u  evidently 

diminishes  with  the  angle  0  and  becomes  nil  when  a  gradually 
and  continuously  curved  bend  is  substituted  for  the  elbow. 

CASE  IV.     Weisbach  gives  the  following  empirical  formula 
for  the  loss  of  head  at  a  bend  in  a  pipe : 


hb  =  mt 


,  d\k 
where  m  =  .131  -f  1.847  — 


for  a  circular  pipe  of  diameter  d,  p  being 
the  radius  of  curvature  of  the  bend,  and 


FIG.  59. 


m  =  .124+  3-104— 


for  a  pipe  of  rectangular  section,  s  being  the  length  of  a  side 
of  the  section  parallel  to  the  radius  of  curvature  (p)  of  the  bend. 
CASE  V.  Valves,  Cocks,  Sluices,  etc. — The  loss  of  head  in 
each  of  the  cases  represented  by  the  several  figures  may  be 
traced  to  a  contraction  of  the  stream  similar  to  the  con- 


FLO W  OF   WATER   IN  PIPES.  93 

traction  which  occurs  in  the  case  of  an  abrupt  change  of  sec- 

v* 
tion.     The  loss  may  be  expressed  in  the  form  m — ,  and  the 

following  tables  give  the  results  obtained  by  Weisbach. 

(a)  Sluice  in  Pipe  of  Rectangular  Section.     (Fig.  60.) — Area 
of  pipe  =  a ;  area  of  sluice  =  s. 


—  =    i      .9 


•5       -4 


.2         .1 


m 


=  .oo  .09   .39  .95   2.08  4.02   8.12  17.8  44.5  193 


FIG.  60. 

(b)  Sluice  in  Cylindrical  Pipe.     (Fig.  61). — s  = 
ratio  of  height  of  opening  to  diameter  of  pipe. 

s=    i     .875    -75    -625      .5       .375       .25      .125 
m  =  .00     .07     .26     .81     2.06     5.52     17.00     97.8 

(c)  Cock  in  Cylindrical  Pipe  (Fig.  62). 

s  =  ratio  of  cross-sections; 

6  =  angle  through  which  cock  is  turned. 


FIG.  61. 


FIG.  62. 


If  0  =     5° 
s  =  ,926 

m  =  .05 

If/=  40° 


.85 

.29 

45° 


=  •385      .315 
=  17.3       31.2 


15° 

.772 

•75 

50° 
.25 
52.6 


FIG  63. 
20°        25°        30°        35° 

.692        .613        .535        .458 
1-56      3-1         547 


55 


106 


60°  65° 
.137  .091 
206  486 


82° 

00 

oo 


(d)   Throttle-valve  in  Cylindrical  Pipe  (Fig.  63) 
0  —  angle  through  which  valve  is  turned. 


94  HYDRA  ULICS. 

If  61  =5°  10°     15°       20°      25°  30°      35°      40° 

^  =  .24  .52       .90       1.54    2.51  3.91     6.22     10.8 

If  0=45°  50°        55°        60°  65°       70°       90 

#2=18.7  32.6       58.8        118  256 


oo 


CASE  VI.  The  fall  of  free  surface-level,  or  loss  of  head,  due 
to  sudden  changes  of  section,  frictional  resistance,  etc.,  may  be 
graphically  represented  as  in  Fig.  64. 


FIG.  64. 


Let  a  length  of  piping  AE  connect  two  reservoirs,  and  let 
h  be  the  difference  of  surface-level  of  the  water  in  the  reser- 
voirs. 

Let  Llt  rl  be  length  and  radius  of  portion  AB  of  pipe. 

«     T     ~    it        «         «         «       «         «        nr"  "      <( 

/-„  rt  ZJCx 

"    L,,  r,  "        "        "        "      "        "       CD  "       • 

"     L     T     "  "  "  "         "  "         JD fi    4i        " 

"    ut,u9,ut,  ut  be  the  velocities  of  flow  in  AB,  BC,  CD, 
DE,  respectively. 


FLOW   OF    WATER   IN  PIPES.  95 

The  reservoir  opens  abruptly  into  the  pipe  at  A. 

There  is  an  abrupt  change  at  B  from  a  pipe  of  radius  rv.  to 
one  of  radius  r^. 

There  is  an  abrupt  change  at  C  from  a  pipe  of  radius  ra  to 
one  of  radius  ra. 

At  D  the  water  flows  through  an  orifice  of  area  A  in  a  dia- 
phragm. At  E  the  velocity  of  the  water  as  it  enters  the  lower 
reservoir  is  immediately  dissipated  in  eddies  or  vortices. 

Draw  the  horizontal  plane  amnop  at  a  distance  from  the 
water-surface  in  the  upper  reservoir  equal  to  the  head  due  to 
atmospheric  pressure. 

Draw  vertical  lines  at  A,  B,  C,  D,  E.     Take 


ab  =loss  of  head  at  the  entrance  A  =  .49—  -  ; 


=   «     u       tt     due  to  faction  from  A  to  B   =fci-j    ; 

r>  ^g 

r*         \a^  a 

cd=.   "     "       "     due  to  change  of  section  at  B=l-^—  il  -*- 

V  i  I   -%3 

re  —   "     "       "     due  to  friction  from  B  to  C    =^-£a  ; 


=   "     "       "     due  to  change  of  section  at  ^=.316-  ; 

o 

—   "     «       «     due  to  friction  from  C  to  D    =^  .  ^-Z 


"     «       "     due  to  change  of  section  at  D=  (^  -  1)  —  ; 


tk  =   "     "       "     due  to  friction  from  D  to  E   —^-  ^L, ; 

z/2 
kl-=   "     "       "     corresponding  to  u  — -— . 


HYDRA  UL1CS. 


Through  /  draw  a  horizontal  plane  Ix.  This  plane  must 
evidently  be  at  a  distance  from  the  water-surface  in  the  lower 
reservoir  equal  to  the  pressure-head  due  to  the  atmosphere. 

Then  the  total  loss  of  head  =  Ip 


ef  +  gh  +  M  +  0C  +  re  +  sg+  tk, 


i 

,   2g       r  r%  2g        '   r9  2g 


2  3       3 


The  broken  line  abcdefghkl  is  the  hydraulic  gradient. 

7.  Remarks  on  the  Law  of  Resistance.  — Poiseuille's  ex- 
periments on  the  flow  of  water  through  capillary  tubes  showed 
that  the  loss  of  head  was  directly  proportional  to  the  ve- 
locity. 

In  the  case  of  pipes  used  in  ordinary  practice  the  loss  is 
undoubtedly  more  nearly  proportional  to  the  square  of  the 
velocity,  and  must  be  mainly  due  to  the  formation  of  eddies. 
These  eddies,  again,  are  formed  more  or  less  readily  according 
as  the  water  possesses  less  or  greater  viscosity. 


FLO W  OF   WATER  IN  PIPES.  97 

The  experiments  of  Unwin  and  others  have  shown  that  the 
surface  friction  is  diminished  by  about  i<f>  for  every  rise  of  5°  F. 
in  the  temperature,  and  it  is  also  known  that  the  viscosity 
diminishes  as  the  temperature  rises  and  vice  versa.  Reynolds 
has  propounded  a  single  law  of  resistance  to  the  flow  through 
pipes,  which  embraces  the  results  of  Poiseuille  and  of  Darcy, 
and  takes  into  account  the  effects  of  viscosity,  temperature, 
etc.  This  law  may  be  expressed  in  the  form 

Bn        vn 
slope  =  *  =  ____ 

where  d  is  the  diameter  of  the  pipe,  A  =  67,700,000,  B  =  396, 
and  P=  (i  +  .0336^  +  .00022 1/2),  the  units  being  metres  and 
degrees  centigrade  (/). 

Unwin  considers  that  the  index  of  the  diameter  d  is  not 
exactly  3  —  n,  and  should  be  determined  independently.  For  a 
rough  surface  n  —  2,  for  a  smooth  cast-iron  pipe  n  =  1.9,  and 
for  a  lead  pipe  n  =  1.723  ;  a  limitation  which  is  analogous  to 
that  found  by  Froude  in  his  experiments  upon  surface  fric- 
tion. 

Experimenting  with  glass  tubes,  Reynolds  found  for  veloc- 
ities below  a  certain  critical  velocity  given  by  the  formula 


that  the  motion  of  the  water  is  undisturbed,  i.e.,  that  it  was  in 
parallel  stream-lines.  At  and  above  this  critical  velocity  eddies- 
are  formed,  and  the  parallel  stream-line  motion  is  completely 
broken  up  within  a  very  short  distance  from  the  mouth  of  the 
tube. 

In  capillary  tubes—  =  43.79. 
In  ordinary  pipes  —  =  278. 


98 


HYDRA  ULICS. 


8.  Flow  of  Water  in  a  Pipe  of  Varying  Diameter.— 

The  variation  in  the  diameter  is  supposed  to  be  so  gradual 

that  the  fluid  filaments  may  still 
be  assumed  to  flow  in  sensible 
parallel  lines. 

Consider  a  thin  slice  of  the 
moving  fluid,  bounded  by  the 
transverse  sections  AB,  CD,  dis- 
tant s  and  s  -f-  ds,  respectively, 
from  an  origin  on  the  axis  of  the 
pipe. 

FIG.  65.  Let/  be  the  mean  intensity  of 

pressure,  A  the  water  area,  P  the  wetted  perimeter  for  the  sec- 
tion AB. 

Let  these  symbols  become  /  +  dp,  A  +  dA,  P  +  dP,  re- 
spectively, for  the  section  CD. 

Let  z  be  the  height  of  the  C.  of  G.  of  the  section  AB 
above  datum. 

Let  z  -f-  dz  be  the  height  of  the  C.  of  G.  of  the  section  CD 
above  datum. 

Let  «,  u  +  du  be  the  velocities  of  flow  across  the  sections 
AB,  CDy  respectively. 
Then 


The  rate  of  increase  of 
momentum  of  the  slice 
ABCD  in  the  direction  of 
the  axis 


f  momentum    generated    by 
the  effective  forces  acting 

Iupon  the  slice  in  the  same 
direction. 


The  acceleration  in  time  dt  =  —Au .  dt-j-  =  —  Au  .  du. 

g  dt      g 

The  total  pressure  on  AB  =  p  .A,  and  acts  along  the  axis. 
The  total  pressure  on  CD  =  (p  +  dp)  (A  +  dA\  and  acts  along 

the  axis. 
The  total  normal  pressure  on  the  surface  ACBD  of  the  pipe 

=  27t[r-\ J  \p  +  — j  A  C  =  2nrp .  A  C,  very  nearly. 


FLOW  OF   WATER  IN  PIPES.  99 

The  component  of  this  pressure  along  the  axis 
=  2nrpAC .sin  6 
=  2  npr .  dr,  nearly, 
6  being  the  angle  between  AC  and  the  axis. 

Thus  the  total  resultant  pressure  along  the  axis 

=  pA  -  (p -\-dp\A  +  dA)  +  2npr.dr 
=  —  p.dA  —  A.dp-}-  27rpr  .  dr 
=  -A.dp, 
since  A  —  7tr\  and  therefore  dA  =  27tr .  dr. 

The  component  of  the  weight  of  the  slice  along  the  axis 

dA\  I          dA\ 

•  w  sm  i  =  —  \A  H )«/•  dz=  —  iv A  .  dz. 


The  frictional  resistance  =  P.AC.  F(u)  =  P  .  ds  .  F(u),  very 
nearly.     Hence 

wAu  .  du 
—~  —  -  =  —  A  .  dp  —  wA  .  dz  —P.  ds  .  F(u\ 

o 

and  therefore 

dpu.du.   PF(u)  y 


Integrating, 

p   ,u'         CP  F(u}  j 
z  +  w  +  ^+J  A  ~^ds  =  a  Constant 


Then 


100 


HYDRAULICS. 


The  integration  can  be  effected  as  soon  as  the  relation  be- 
tween r  and  s  is  fixed. 

Example.  —  Take  r  =  a  -f-  bst  and  assume  /and  Q  to  be  con- 
stant. Then 

£_L"C__L  —  +  -r  —  3   /  —  =  a  constant, 
'    w~  2g~  b  gn  J   r" 


and  therefore 


z  i  <_  _|_  —   i    _  —  _  —  —  a  constant. 

'    w      2      '          -2     4 


9.  Equivalent  Uniform  Main.  —  A  water-main  usually  con- 
sists of  a  series  of  lengths  of  different  diameters. 

As  a  first  approximation  the  smaller  losses  of  head  due  to 
changes  of  section,  etc.,  may  be  disregarded,  and  the  calcula- 
tions may  be  further  simplified  by  substituting  for  the  several 
lengths  a  single  pipe  of  uniform  diameter  giving  the  same  fric- 
tional  loss  of  head.  Such  a  pipe  is  called  an  equivalent  main. 


FIG.  66. 

Let  /,,/„,  /3    be  the  successive  lengths  of  the  main. 
Let  dt  ,  d^  ,  d^  be  the  diameters  of  these  lengths. 
Let  z/v,  vt,  v3  be  the  velocities  of  flow  in  these  lengths. 
Let  //,  ,  h^  ,  h^  be  the  frictional  losses  of  head  in  these  lengths. 
Let  Z,,  d,  v,  h  be    the    corresponding     quantities    for    the 
equivalent  uniform  main. 
Then 

h  =  //,  +  h,  +  h,  +  .  .  .  , 

and  therefore 


r  _  ,    ,  ,    , 

~~          ~  1 


FLOW  OF  WATER  IN  PIPES.  IOI 

Hence 


where  it  is  assumed  that  /is  the  same  for  the  several  lengths 
of  the  main  and  also  for  the  equivalent  pipe. 
But 

nd*  nd?  nd? 

TV  =  Q  =  —Vi  =  —v,=  «c. 

Hence 

L        I,        /2         /3 


an  equation  giving,  the  length  L  of  an  equivalent  pipe  having 
the  same  total  frictional  loss  of  head. 

10.  Branch  Main  of  Uniform  Diameter.  —  In  a  branch  main 
AB  of  length  L  and  diameter  d,  receiving  its  supply  at  A.  — 
Let  Qw  be  the  way-service,  i.e.,  the  amount  of  water  given 

up  to  the  service-pipes  on  each  side. 
Let  Q  be  the  end-service  i.e.,  the  amount  of  water  dis- 

charged at  the  end  B. 
Then  it  may  be  assumed,  and  it  is  approximately  true,  that 

the  way-service  per  lineal  foot,  viz.,  -JT-,  is  constant. 

Thus  the  amount  of  water  consumed  in  way-service  in  a 
length  AC  of  the  main,  where  BC  =  s,  is 


while  the  total  amount  of  water  flowing  across  the  section  of 
the  pipe  at  C 


•v  being  the  velocity  of  flow  at  C. 


I O2  H  YDRA  ULICS. 

Now  dh,  the  frictional  loss  of  head  at  C  for  an  elementary 
length  ds  of  the  pipe,  is  given  by  the  equation 


=  32. 
Integrating,  the  total  loss  of  head  is 


SPECIAL  CASES. 


CASE  I.  Let  <2/  be  the  total  discharge  for  the  same  fric- 
lional  loss  of  head,  ^,  when  the  whole  of  the  way-service  is 
stopped.  Then 


or  =  &•  +  Q.Q.  +  Qjf- 

0 


and  therefore 


Hence 


and  <2/  lies  between  g«  +  —  and  Qe+  —  7=QW,  its  mean  value 

V  3 


being  & 

CASE  II.     If  there  is  no  end-service,  all  the  water  having 

been  absorbed  in  way-service,  Qe  =  o,  and  therefore  Q'e  =  —r= 

V  $ 
and 


FLOW   OF   WATER   IN  PIPES.  1  03 

CASE  III.    If  Qe  =  o, 

fQ™ 
dh  =     vTsT^ds  ==  elementary  f  fictional  loss  of  head. 

Integrating  between  o  and  s, 


and  the  vertical  slope,  or  line  of  free  pressure,  becomes  a  cubical 
parabola. 

CASE  IV.  Let  the  main  receive  its  supply  at  A  from  a 
reservoir  X  in  which  the  surface  of  the  water  is  hl  above  datum, 
and  let  it  discharge  at  the  end  B  into  a  reservoir  Fwith  its 
surface  I?  above  datum. 

Since  (QeJ  =  Q?  +  0CQW  +  ^,  therefore 


If  Qw  =  TsQ.',  Qe  =  o;  and  if  Qw  >  3g/,  then  the  res- 
ervoir  Fwill  furnish  a  portion  of  the  way-service. 

Suppose  that  X  gives  the  supply  for  the  distance  AO 
(=  /,)  and  that  Ksupplies  BO  (=  /a). 

Let  z  be  the  height  above  datum  of  the  surface  in  a  press- 
ure column  inserted  at  O. 

Then,  neglecting  the  loss  of  head  at  entrance, 


w 


i  fQ  V" 

=  loss  of  head  between  A  and  O  =  —    r» /a, 

3  ^  a  L, 

and 


J       fQ     2/S 

=  loss  of  head  between  B  and  O  = rybi 

3  n  d  L 


Also  A  +/,  =  L. 


104 


HYDRA  ULICS. 


II.  Nozzles. — Let  a  pipe  AB,  of  length  /  and  diameter  d, 
lead  from  a  reservoir  h  ft.  above  the  end  B. 

First,  let  the  pipe  be  open  to  the  atmosphere  at  B. 


FIG.  68. 


Then 


(v* 
=  n — 
2g 

I 

-f-  head  to  overcome  resistance  due  to  bends,  etc.    =  m— 

V         2 

•4-  head  to  overcome  frictional  resistance  (=  — >) 

\       d  2gl 

-|-  head  corresponding  to  the  velocity  v  in  the  pipe  and  at 
the  outlet  f=  *  J 


4/A    I    ^ 
d  }       2g 


Hence  the  height  to  which  the  water  is  capable  of  rising 
B 

v\ 


or,  again,  is 


=f=A-- 

h 


4/ 

^-t 

d  r 


£ 

~d 

Second,  let  a  nozzle  be  fitted  on  the  pipe  at  B. 
Let   V  be  the  velocity  with  which  the  water  leaves  the 
nozzle. 


FLO  W  OF   WATER   IN  PIPES.  1  05 

Let  D  be  the  diameter  of  the  nozzle-outlet. 
This  diameter  is  very  small  as  compared  with  the  diameter 
d  of  the  pipe.     But 


T7 

V  =  —  v, 


4  4 

and  therefore 


so  that  Fis  very  large  as  compared  with  i>. 
Also, 

h  =  head  to  overcome  the  resistance  to  entrance  at  A 

-\-  head  to  overcome  the  resistance  due  to  bends,  etc. 

-f-  head  to  overcome  the  frictional  resistance  in  pipe 

+  head  to  overcome  the  frictional   resistance   in    nozzle 

(=*•£) 

V  2g ) 

-f-  head  corresponding  to  the  velocity  V  with  which  the 

/      V**\ 
water  leaves  the  nozzle    —  — 


,   4/A          ,F3   .    V 
—  — n  -f  m  +  ^-    +  m'—  +  — , 

2g\  dl  2g^  2g 

and  the  height  to  which  the  water  is  now  capable  of  rising  at 
j5is 

v*       7       v*(     .         .  4/A          ,Fa 
—  =  h  —  — (n-\-  m-4-  ^—}  —  m' — 

2g  2g\  d  I  2g 

h 


Let  —  ,  =  //„,  be  the  pressure-head  at  the  entrance  to  the 
w 

nozzle.     Then  the  effective  head  at  the  same  point 


Hence 


io6 


HYDRAULICS. 


It  will  be  observed  that  the  delivery  from  the  nozzle  is  less 
than  that  from  the  pipe  before  the  nozzle  was  attached,  but 
that  the  velocity-head  at  the  nozzle-outlet  is  enormously  in- 
creased. The  actual  height  to  which  the  water  rises  on  leav- 
ing a  nozzle  is  less  than  the  calculated  height,  owing  to  air- 
resistance  and  to  the  impact  of  particles  of  water  as  they  fall 
back. 

The  force  required  to  hold  the  nozzle  is  evidently 


g  £4 

If  the  water  flowing  through  a  pipe,  or  hose,  of  length  /  ft., 
with  a  velocity  of  v  ft.  per  second,  is  quickly  and  uniformly 
shut  off  by  a  stop-valve  t  sec.,  the  pressure  in  the  pipe  near  the 

valve  is  increased  by  an  amount  -    -  Ibs.  per  square  foot. 

<3 

Of  two  forms  of  nozzle  in  general  use,  the  one  (Fig.  70)  is  a 


FIG.  69.  FIG.  70. 

surface  of  revolution  with  a  section  which  gradually  diminishes 
to  the  outlet,  while  the  other  (Fig.  69)  is  a  frustum  of  a  cone, 
having  a  diaphragm  with  a  small  circular  orifice  at  the  outlet. 
Denoting  the  former  by  A  and  the  latter  by  £,  the  following- 
table  gives  the  results  of  Ellis's  experiments : 


Height  of  jet  from 
i-inch  Nozzle. 

Height  of  jet  from 
i£-mch  Nozzle. 

Height  of  jet  from 
iHnch  Nozzle. 

Pressure  in  Ibs. 

Head  in 

per  sq.  in. 

feet. 

A 

B 

A 

B 

A 

B 

10 

23 

22 

22 

22 

22 

23 

22 

20 

46 

43 

42 

43 

43 

43 

43 

30 

69 

62 

61 

63 

62 

63 

63 

40 

92 

79 

78 

81 

79 

82 

80 

50 

"5 

94 

92 

97 

94 

99 

95 

60     , 

138 

108 

104 

112 

108 

"5 

no 

70 

161 

121 

"5 

125 

121 

129 

123 

Sc 

184 

131 

124 

137 

131 

142 

135 

90 

207 

140 

132 

148 

141 

154 

146 

•         IOO 

230 

148 

136 

157 

149 

164 

155 

FLOW  OF   WATER   IN  PIPES.  IO/ 

Third,  if  an  engine,  working  against  a  pressure  of  pc  Ibs.  per 
square  foot,  pumps  Q  cubic  feet  of  water  per  second  through 
a  nozzle  at  the  end  of  a  hose  /  feet  in  length,  then 

the  pumping  H.P.  of  the  engine  ==  —  —  . 

The  total  head  at  the  engine  end  of  the  hose  =  the  head 
corresponding  to  the  pressure  p  in  the  hose  -f~  the  head  re- 
quired to  produce  the  velocity  of  flow  v 


W          2g 

and  this  head  is  expended  in  overcoming  the  frictional  resist- 
ance of  the  hose  (all  other  resistances  are  disregarded)  and  in 
producing  the  velocity  of  flow  Fat  the  outlet.  Hence 


W  W  2g  d     2g  2g 

and  therefore 


W  d     2g          2g 


-  _       JL 

gn*  * 


.-  Ttd  41.J-S      T- 

since  Q  =  v  = F. 

4  4 


The  pumping  H.P. 

8wff_(j_        4/7\ 
-o7t*\D*          d*  r 


12.    Motor  Driven  by  Water  from  a  Pipe.  —  Let  the 

nozzle  in  the  preceding  article  be  replaced  by  a  cylinder  hav- 
ing its  piston  driven  by  the  water  from  the  pipe. 

Let  u  =  the  velocity  of  the  piston  per  second. 

Let  pm  =  unit  pressure  at  the  end  of  the  pipe,  i.e.,  in  the 
cylinder. 

Let  dm  —  diameter  of  cylinder. 


IO8  HYDRA  ULICS. 

Then,  velocity  of  flow  in  pipe  =  ~jp-u*     Hence 
,  _    d^     ul     ,     4//   dm<    u*        pm 

(other  losses  of  head  being  disregarded). 

13.  Siphons. — A  siphon  is  a  bent  tube,  ABCD,  Fig.  71,  and 
>-— r—  is  often  employed   to  convey 
water  from  one    reservoir   to 
another  at  a  lower  level. 

Let  hv  //3,  respectively,  be 
the  differences  of  level  be- 
tween the  top  of  the  siphon 
and  the  entrance  A  and  outlet 
D  to  the  siphon.  Then,  so 
long  as  the  height  k1  does  not 
exceed  the  head  of  water 
(=  32.8  ft.)  which  measures 
the  atmospheric  pressure,  the 
FlG*  7I>  water  will  flow  along  the  tube 

in  the  direction  of  the  arrow,  with  a  velocity  v  given  by  the 
equation 


/being  the  length  of  the  tube  ABCD,  and  all  resistances,  ex- 
.cept  that  due  to  frictional  resistance,  being  disregarded. 

If  //,  >  32.8  feet,  each  of  the  branches  AB  and  DC  becomes 
a  water-barometer,  and  the  siphon  will  no  longer  work. 

Even  when  the  siphon  does  work,  an  arrangement  must 
be  made  for  withdrawing  the  air  which  will  always  collect  at 
the  upper  part  of  the  siphon, 

14.  Inverted  Siphons. — The  existence  of  a  cutting  or  a 
valley  sometimes  renders  it  necessary  to  convey  the  water 
from  a  course  AB  to  a  course  DE  by  means  of  an  inverted 
siphon  BCD  of  length. 

Let  u  be  the  velocity  of  flow  in  AB,  and  h  the  height  of  B 
above  a  datum  line. 


FLOW  OF   WATER   IN  PIPES,  1 09 

Let  v  be  the  velocity  of  flow  in  the  siphon,  and  ht  the  height 
of  D  above  datum. 


FIG.  72. 
Then 

h^  —  7za  =  loss  of  head  at  B 

-\-  frictional  loss  of  head  in  siphon 
loss  of  head  at  D 


=          , 

zg        d    2g  "•"  2g 

4/7  v* 
=  ZL  --  ,  approximately, 

assuming  the  entrance  and  outlet   to   the  siphon    formed   in 

u*  v* 

such  a  manner  as  to  considerably  reduce  the  losses  —  and  —  , 

zg          2g 

and  to  allow  of  these  losses  being  disregarded  without  practical 
error.  Find,  by  chaining  along  the  ground,  the  length  of  the 
siphon  from  B  up  to  a  point  F  not  far  from  D.  Call  this 
length  /,  ,  and  let  \  be  the  height  above  datum  of  F,  obtained 
with  a  level.  Generally  speaking,  DF  is  nearly  always  of 
uniform  slope.  Call  the  slope  a.  Then, 

DF  =  (k^  —  h^)  cosec  a. 
But 


=  hl  —  h^  —  DF.  sin  #, 

an  equation  from  which  DF  can  be  found,  as  /^  —  h^  can  be 
determined  by  means  of  a  level. 


10 


HYDXA  ULICS. 


15.  Air  in  a  Pipe. — The  effect  of  an  air-bubble  in  a  pipe 
ABCD  may  be  discussed  as  follows: 

Let  the  air  occupy  the  portion  BC  of  a  pipe. 

Let  the  surface  of  the  water  in  the  reservoir  supplying  the 
pipe  be  h^  ft.  vertically  above  E,  and  hz  ft.  above  D. 


FIG.  73. 

Also,  let  h^  be  the  difference  of  level  between  C  and  D,  ht 
the  difference  of  level  between  B  and  C,  and  /  the  thickness  of 
the  water-layer  EF. 

Let  H  designate  the  head  equivalent  to  the  elastic  resist- 
ance of  the  air  in  BC.  Then,  approximately, 


and 


A 


/!  being  length  of  portion  of  pipe  from  A  to  E,  and  /,  the  length 
from  E  to  D. 

Adding  the  two  equations, 

L    IL        f  -  4/  *>*  ,,    ,    ...  _  4//  v* 

/*,+;,,-/-.  __(/l  +  /,).  __, 

/  being  total  length  of  pipe. 

But  //!  —  /  +  h<  =  //,  —  h^  ,  very  nearly.     Hence 


an  equation  showing  the  variation  of  v  with  a  variation  in  the 
height  /*4  of  the  space  occupied  by  the  air. 

Note.  —  H  o>{  course  varies  with  the  temperature. 


FLOW  OF   WATER  IN  PIPES. 


Ill 


16.  Three  Reservoirs  at  Different  Levels  connected  by 
a  Branched  Pipe. — Let  a  pipe  DO  of  length  /x  ft.  and  radius 
rl  ft.,  leading  from  a  reservoir  A  in  which  the  water  stands  hl 
ft.  above  datum,  divide  at  O  into  two  branches,  the  one,  OE, 
of  length  /2  ft.  and  radius  r2  ft.,  leading  to  a  reservoir  B  in 
which  the  water  stands  //2  ft.  above  datum,  the  other,  OF,  of 
length  /3  ft.  and  radius  r3  ft.,  leading  to  a  reservoir  C  in  which 
the  water  stands  h  ft.  above  datum. 


I 


-. 


FIG.  74. 

Let  vlt  v^  vz  be  the  velocities  of  flow  in  DO,  OE,  OF,  re- 
spectively. 
Let  Qlt  <2a,  Q,  be  the  quantities  of  flow  in  DO,  OE,  OF, 

respectively. 
Let  z  be   the   height   above   datum   to  which   the  water 

will  rise  in  a  tube  inserted  at  the  junction. 
Two  problems  will  be  considered,  and  all  losses  of   head 
excepting    those   due  to    frictional   resistance  will  be   disre- 
garded. 

PROBLEM  I.     Given  h,,  h^  hz ;  rlf  r9,  r3 ;  to  find  Qlf  <23,  Q3 1 
•z.', ,  z>2,  i>3,  and  -S". 

fo ^       ^  a 

For  the  pipe  DO,     -~—  =  a-    .  .  (i)   and    Q^Ttrfv,.    .  .  (2) 
**  ^\ 


=«V     .  .  (3)     "      Q.=  «r:Vf      •  (4) 


112  HYDRAULICS. 

For  the  pipe  OF,     Z-^-^=a^*-   .  .  (5)   and    Q3=nr3*vs.    .  .  (6) 

Also,  01=  ±<2,+  <2..      •    V.;V    ;  ,    .    (7) 

From  these  seven  equations  the  seven  required  quantities 
can  be  found. 

In  equations  (3)  and  (7),  the  upper  or  lower  signs  are  to  be 
taken  according  as  the  flow  is  from  O  towards  £  or  from  R 
towards  O. 

This  may  be  easily  determined  as  follows  : 

Assume  z  —  7za,  and  then  find  vl  and  v^  by  means  of  equa- 
tions (i)  and  (5),  and  hence  Q^  and  <23  by  means  of  equations 
(2)  and  (6).  If  it  is  found  that  Ql  >  Q9,  then  the  flow  is  from 
O  to  E,  and  equations  (3)  and  (7)  become 

s=«^  and     = 


while  if  <2,  <  <23,  the  flow  is  from  E  to  O,  and  the  equations 


are 

-?— —  =  <*—     and 


(-9- 


. — It  is   assumed  that  a   —  - is  the  same  for  each 

pipe. 

SPECIAL  CASE.    Fig.  75. — Suppose  the  pipe  OE  closed  at  E. 

Also  let  r^  =  ra  =  ra  =  r,  and  let  V  be  the  velocity  of  flow 
from  A  to  C. 

The  "  plane  of  charge  "  for  the  reservoir  A  is  a  horizontal 

plane  MQ  distant  —  from  the  water  surface,  /0  being  the  at- 
mospheric pressure. 

The  "  plane  of  charge  "  for  the  reservoir  C  is  a  horizontal 

plane  TS  distant  —  from  the  water-surface. 
w 

Fa 

In  the  vertical  line    VTQ,  take   TN  —  —   and  join  MN. 

2g' 

Then,   neglecting   the  loss   of  head   at   entrance,  MN  is  the 


FLOW  OF   WATER   IN  PIPES.  113 

"  line  of  charge,"  or  hydraulic  gradient,  for  the  pipe  DF,  and 
is  approximately  a  straight  line. 

Let  the  "  plane  of  charge  "  KK  for  the  reservoir  B,  distant 

—  from  the  water-surface,  meet  MN  in  G. 

If  the  junction  O  is  vertically  below  G,  there  is  no  head 
._£--         —  £ £ _<?___ 


1 

I 

» 

! 

Ps, 

, 
• 

•  if* 

^j._  —  :>, 

^  ... 

7^ 

>i 

vvv 

7^^ 

°x--* 

tSJnfa 

TIT 

\^ 

f\-    - 

m. 

f    3 

KTxv 

^^ 

/ 

x\ 

r*4 

, 

f\ 

;s 

i   "   / 
V                                       m 

f 

i 


FIG.  75- 

available  for  producing  flow  either  from  E  towards  O  or  from 
O  towards  E,  and  hydrostatic  equilibrium  is  established. 

If  the  junction  O  is  on  the  left  of  G,  and  a  vertical  line 
OKHL  is  drawn  intersecting  KK,  MN,  and  MQ  in  the  points 
K,  H,  and  L,  there  is  the  head  Hf£  available  for  producing 
flow  from  O  towards  E. 

If  the  junction  O  is  on  the  right  of  G,  and  the  vertical  line 
OHKL  is  drawn,  the  head  HK  is  now  available  for  producing 
flow  from  E  towards  O. 

Let  the  vertical  through  G  meet  MQ  in  Pt  and  take 
PG  =  Y.  Then,  approximately, 


I,  +  /  ~~  MN~  QN~  h,- 


H4  HYDRA  ULICS. 

and  therefore 

y  -  k*  ~  k*    j 

'      A+A      '       '' 

If  HL  <  F,  the  flow  is  from  O  towards  £. 
If  HL  >  F,  "  "  "  "  E  "  O. 
Again, 


w       2gl  r 

and  therefore,  approximately, 


Next  assume  the  junction  O  to  be  on  the  left  of  G,  and 
open  the  valve  at  E.     Then 


and        Q,=  Q,+  Q,, 
or  z/t  =  v,  +  vt. 

Thus 

«y(A+  O  =  *,-*,  =  "(/^,1+  A*,*)  =  "  j  A(».+  fJ'+Af  ,'  }  ; 


and  therefore 

^.'(A  +  A)  +  2/w,  +  A^.1  -  (A  +  A)  ^  =  o. 

Hence,  assuming  z/a  very  small  as  compared  with  V, 


or 


where  Q  =  nr*  V. 


FLOW  OF   WATER   IN  PIPES.  115 

Thus  it  appears  that  if  a  quantity  <23  of  water  is  drawn  off 
by  means  of  a  branch  from  a  main  capable  of  giving  a  total 
end  service  Q,  this  end  service  will  be  diminished  by  j-<22,  \Q^ 
\Qv  etc.  according  as  the  junction  O  divides  the  pipe  DF  into 
two  portions  in  the  ratio  of  I  to  I,  I  to  2,  I  to  3,  etc. 

Note. — The  more  correct  value  of  v^  is 


/,+/ 


and  the  maximum  value  of  — — :-LTT-«  does  not  exceed  — . 

4 


Orifice  Fed  by  Two  Reservoirs. — Neglect  all  losses  of  head 
except  the  losses  due  to  frictional  resistance. 


FIG.  76. 


When  the  valve  at  0  is  closed  the  flow  is  wholly  from  A  to 
and  the  delivery  is 


The  line  of  charge  (hydraulic  gradient)  is  -M/V,  where 


.  w 


Il6  HYDRAULICS. 

Open  the  valve  a  little  :  a  volume  <2a  will  now  flow  through 
<9,  and  a  volume  (23  mto  £  where 


The  "  line  of  charge"  becomes  the  broken  line  MiN. 

As  the  opening  of  the  valve  continues,  the  pressure-head  at 

O  diminishes,  and  when  it  is  equal  to  /z3  +  —  °  the  line  of  charge 

\sM2N,  2  N  being  horizontal.  Hydrostatic  equilibrium  is  now 
established  between  O  and  C,  and  the  whole  of  the  water  from 
A  passes  through  O,  the  delivery  being  given  by 


Opening  O  still  further,  both  reservoirs  will  serve  the  ori- 
fice, and  the  line  of  charge  will  continue  to  fall. 

When  the  valve  is  full  open  the  "line  of  charge"  is 

where  3(9  =  —  ,  and  the  discharge  is 


w 


The  supply  from  A  is  equal  to  that  from  C  when  -1  =  — *. 

The  above  investigation  shows  the  advantage  of  a  second 
reservoir  in  emergent  cases  when  an  excessive  supply  is  sud- 
denly demanded,  as,  e.g.,  on  the  occasion  of  a  fire. 

PROBLEM  II.  Given  /z,,  //2,  h^\  Q2,  Qs,  and  therefore 
Qt(=  ±  a+G3);  tofirtdrI,r;,rt,f>l,9t,f'.,jr. 

As  before,  let  z  be  the  pressure-head  at  O.     Then 

...     (i)     and     <2,  =  «•>,;    ...     (2) 

•••     (3)       "       e,  =  *r,'Vt;    ...     (4) 
...     (5)       "       C.  =  ^>..     .     .     .     (6> 


FLOW  OF   WATER  IN  PIPES.  1 1/ 

These  six  equations  contain  the  seven  required  quantities, 
viz.,  r1 ,  ra ,  r5 ,  z\  ,  v9,  vt,  and  z.  Thus  a  seventh  equation 
must  be  obtained  before  their  values  can  be  found.  This 
equation  is  given  by  the  condition  "  that  the  cost  of  the  piping 
laid  in  place  should  be  a  minimum/'  it  being  assumed  that  the 
cost  of  a  pipe  laid  in  place  is  proportional  to  its  diameter. 

Hence 

llrl  +  4ra  +  4ra  —  a  minimum (7) 

From  equations  (i)  and  (2),       -L— —  —  — j-^; 


(3)         (4),  -  -^ 
"         (5)    "    (6),       *-^b  = 


^3          rr, 

Differentiating  these  three  equations, 
dz  _  $aQ*     , 


t 


But  by  equation  (7) 

/X^  -|_  l^dr^  +  /3^r3  =  O. 
Hence 


6  „  6 


which  is  the  seventh  equation  required. 


Il8  HYDRAULICS. 

This  last  equation  may  be  written  in  the  forms 


and 

a  =  ±  a  v  a 

^3  z/,1       ^33" 

17.  Mains  with  any  Required  Number  of  Branches. 

Let  there  be  n  junctions  and  m  pipes. 

Let  hl ,  //a ,  .  .  .  hm  be  the  m  pressure-heads  at   the  end  of 

each  successive  length  of  pipe. 
Let  #!,£,,...#„  be  the  n  pressure-heads  at  the    1st,  2dy 

3d,  .  .  .  72th  junctions. 

Let  /!,/,,.../,„  be  the  lengths  of  the  ;//  pipes. 
PROBLEM  I.  Given  //, ,  h^ ,  .  .  .  hm  ,    rl ,  r2 ,  .  .  .  rm ,    to    find 

?i;tf9,.  *•»»,*, »*,».••**«• 

_i.  ^  nz  £          £>2 
There  are  772  equations  of  the  type  -  -——  -  =  a—. 

Also,  the  quantity  flowing  through  the  first  portion  of  the 
main  is  equal  to  the  sum  of  the  quantities  flowing  through  all 
the  branches  at  the  first  junction,  and  an  analogous  equation 
will  hold  for  each  of  the  remaining  n  —  I  junctions.  Thus  n 
additional  equations  are  obtained. 

From  these  m  -f-  n  equations,  vl ,  vz ,  .  .  .  vm ,  z^ ,  ^ ,  .  .  .  zn 
may  be  found  analytically  or  by  the  method  of  repeated  ap- 
proximation. 

PROBLEM  II.   Given  h^ ,  h^ ,  .  .  .  hm ,  Ql ,  <22 ,  .  .  .  Qm ,  to  find 

There  are  now  only  m  equations  of  the  type 

-[-  h  If  %  _       V* 

~T~         a~r  ' 

involving  m  -\-  n   unknown  quantities,  and  the  problem  admits 
of  an  infinite  number  of  solutions. 

It  is  therefore  assumed  that  the  cost  of  the  piping  laid 
in  place  is  to  be  a  minimum.  Thus  n  new  equations  are  ob- 


FLO W  OF   WATER  IN  PIPES. 

tained,  and  the  m  -\-  n  equations  may  be  solved  analytically  or 
by  repeated  trial. 

18.  Variation  of  Velocity  in  a  Transverse  Section.— 
Assumption. — That  the  water  in  any  portion  of  a  pipe  is  made 
up  of  an  infinite  number  of  hollow  concen- 
tric cylinders  of  fluid,  each  moving  parallel 
to  the  axis  with  a  certain  definite  velocity. 

Let  u  be  the  velocity  of  one  of  these  cyl- 
inders of  radius  x  and  thickness  dx.  Then 
the  flow  across  a  transverse  section  is  given 
by  the  equation  FlG  ?7 

dq  =  2nx  dx  .  u, 
and  the  total  flow 

Q—27tl   uxdx, (i) 


r  being  the  radius  of  the  pipe. 

If  vm  be  the  mean  velocity  for  the  whole  transverse  section 
of  the  pipe, 


nr* 


(2) 


Again,  assuming  with  Navier  that  the   surface  resistance 
between  two  concentric  cylinders  is  of  the  nature  of  a  viscous 

resistance  and  may  be  represented  by  k—  per  unit  of  area  at 

dx 

the  radius  x,  k  being  a  coefficient  called  the  coefficient  of  vis- 
cosity, then  the  total  resistance  at  the  radius  x  for  a  length  ds 
of  the  cylinder 

,  du  du 

=  —  2nx  .  ds  .  k  —  -  =  —  2nk  .  as  .  x--. 

dx  dx 

The  total  resistance  at  the  radius  x  -f-  dx 
du    ,     < 


I2O  HYDRAULICS. 

Hence  the  total  resultant  resistance  for  the  length  ds  of  the 
cylinder  under  consideration 

=  2nkds  — 

The  component  of  the  weight  of  the  slice  of  the  cylinder 
in  the  direction  of  the  axis 

=  w  .  2nx .  dx .  ds  .  sin  0, 

0  being  the  inclination  of  the  axis  to  the  horizon. 

Let  —  dz  be  the  fall  of  level  in  the  distance  ds.     Then 

—  dz  =  ds  .  sin  6. 
Therefore,  component  of  weight  in  direction  of  axis 

=  —  w  .  2nx  dx .  dz. 
The  resultant  pressure  on  the  slice  in  the  direction  of  motion 

=  P  —  (P  ~\~  d£) .  znx .  dx  =  2nx  .dx  .dp. 
Then,  since  the  motion  is  uniform, 

w  .  2nk  .  ds  .  —r\x—-\dx  —  w  .  2rtx  .dx.dz  —  2nx  .dx  .dp  —  o, 
dx\  dxi 

and  therefore 

k .  ds  d  f  du\  dp 

/t*  _    ,/V ty f.     


-T-U-H  -  dz  -  -£  =  o. 

x     ax\  ax  I  w 


Integrating  only  for  the  cylinder  under  consideration, 

ks  d  f  du\       (         p\ 

--  —\x-r]  —  (z  +  —  )  =  a  constant. 

x  ax\  ax  I        \         w' 

But  z  +  —  is  evidently  independent  of  x,  and  is  a  linear 
w 

function  of  s  (Art.  2,  Chap.  III.).     Hence 

I  d  I  du\ 

--  (x—    =  a  constant  =  A,  suppose. 

x  dx\  dx> 


FLO  W   OF   WATER  IN  PIPES.  121 

Therefore 

d  I  du\ 

-(,-J  =  ^..    .    .    .    .    .  ..    (3) 

Integrating, 

du          x* 

x—-  =  A \-  B. 

dx  2 

Assuming  that  the  central  fluid  filament  is  the  filament  of 
maximum  velocity,  then  when  x  =  o,  —  -  is  also  nil.     Therefore 


B  =  o     and     x^  =  Ax\ 
dx 


and  therefore 
Integrating, 


u  =  A-  +  C. 
4 


Let  wmax  be  the  velocity  of  the  central  filament,  i.e.,  the 
value  of  u  when  x  =  o. 
Then 


(5) 

where  D  = . 

4 
Again,  by  equation  I, 

Q  =  27tJ  (#max  —  Dx^x.dx  =  7rr*\2£max — J : 

and  by  equation  2, 

Dr* 
vm  =  umax (6) 

2 

If  ;/,  =  surface  velocity,  then,  by  equation  5, 

U,  =  «max  -  DS (7) 

Hence,  by  equations  6  and  7, 

us  +  «inax  —  zvm (8) 


122 


i  v  ,-r  y 
U 

•  '$*£' 

V-M         '•    f5   T 

:  5:< 

*^ 

c 

( 

-  ^ 

ULICS. 

' 

-1 

-^ 

I 

^\: 

i^ 

EXAMPLES. 


r  sec.  / 

4  ft.  in    J 


1.  A  water-main  is  to  be  laid  with  a  virtual  slope  of  i  in  850,  and  is  to 
give  a  maximum  discharge  of  35  cubic  feet  per  second.     Determine  the 
requisite  diameter  of  pipe  and  the  maximum  velocity,  taking/  =  .0064.    Q  V- 

Ans.  3.679  ft.;  3.2888  ft.  per  sec. 

2.  Find  the  loss  of  head  due  to  friction  in  a  pipe  :  diameter  of  pipe        / 
=  12  in.,  length  of  pipe  =  5280  ft.,  velocity  of  flow  =  3  ft.  per  second  ;    *^ 
f  =  .0064.;     Also  find  the  discharge. 

Ans.   19.008  ft.  ;  2.3562  cub.  ft.  per 

3.  A  pipe  has  a  fall  of  10  ft.  per  mile  ;  it  is  10  miles  long  and 
diameter.     Find  the  discharge,  assuming/  =  .0064. 

Ans.   54.7  cub.  ft.  per  sec. 

4.  A  pipe  discharges  250  gallons  per  minute  and  the  head  lost  in  fric- 
tion is  3  ft..     Find  approximately  the  head  lost  when  the  discharge  is  300 
gallons  per  minute  ;  also  find  the  work  consumed  by  friction  in  both 
cases.  Ans.  4.32(1.;  7500  ft.-lbs.  ;  12,960  ft.-lbs. 

5.  What  is  the  mean  hydraulic  depth  fn1  a  circular  pipe  when  the 

diameter 
water  rises  to  the  height  --  -=-  above  the  centre  ? 

2^2  I0 

Ans.  —  x  diameter. 

6.  A  12-inch  pipe  has  a  slope  of  12  feet  per  mile;  find  the  discharge. 
(/=.oo5.)  Ans.   2.  ii^  cub.  ft.  per  sec. 

7.  The  mean  velocity  of  flow  in  a  24-in.  pipe  is  5  ft.  per  second  ;  find 
its  virtual  slope,/  being  .0064.  Ans.  i  in  200. 

8.  Calculate  the  discharge  per  minute  from  a  24-in.  pipe  of  4000  ft, 
length  under  a  head  of  80  ft.,  using  a  coefficient  suitable  for  a  clean  iron 
pipe.  Ans.  34.909  cub.  ft.  per  sec. 

9.  How  long  does  it  take  to  empty  a  dock,  whose  depth  is  31  ft,  6 
ins.  and  which  has  a  horizontal  sectional  area  of  550,000  sq.  ft.,  through 
two  7-ft.  circular  pipes  50  ft.  long,  taking  into  account  resistance  at  en- 
trance ?  Ans.  214  min.  6  sec. 

10.  The  virtual  slope  of  a  pipe  is  i  in  700;  the  delivery  is  180  cubic 
-feet  per  minute.     Find  the  diameter  and  velocity  of  flow. 

Ans.   1.26  ft.;  2.401  ft.  per  sec. 

n.  Determine  the  diameter  of  a  clean  iron  pipe,  100  feet  in  length, 
which  is  to  deliver  .5  cub.  ft.  of  water  per  second  under  a  head  of  5  feet.. 
Assume/  =  .006.  Ans.  .326ft. 


FLOW  OF   WATER   IN  PIPES.  123 

12.  A  reservoir  has  a  superficial  area  of  12,000  ft.  and  a  depth  of  60 
ft. ;  it  is  emptied  in  60  minutes  through  four  horizontal  circular  pipes, 
equal  in  diameter  and  50  ft.  long.     Find  the  diameter.       Ans.  1.75  ft. 

Explain  how  the  total  head  is  made  up,  and  draw  the  plane  of  charge. 

13.  A  3-inch  pipe  is  very  gradually  reduced  to  i  inch.     If  the  press- 
ure-head in  the  pipe  is  40  ft.,  find  the  greatest  velocity  with  which  the 
water  can  flow  through.  Ans.   1.4  ft.  per  sec. 

14.  Water  flows  through  a  24-inch  pipe  5000  yards  in  length.     At  1000 
yards  it  yields  up  300  cubic  feet  per  minute  to  a  branch.     At  2800  yards 
it  yields  up  400  cubic  feet  per  minute  to  a  second  branch.     At  4000  yards 
it  yields  up  600  cubic  feet  per  minute  to  a  third  branch.     The  delivery  at 
the  end  is  500  cubic  feet  per  minute.    Find  the  head  absorbed  by  friction. 
(/=.oo75.)  Ans.  176.801  ft. 

1 5.  Find  the  H.  P.  required  to  raise  550  gallons  per  minute  to  a  height 
of  60  feet,  through  a  pipe  100  feet  in  length  and  6  in.  in  diameter,  the 
coefficient  of  friction  being  .0064.  ,  Ans.  10.74. 

1 6.  What  head  of  water  is  required  for  a  $-in.  pipe,  150  ft.  in  length,, 
to  carry  off  25  cub.  ft.  of  water  per  minute  ?  Ans.  1.56223  ft. 

What  head  will  be  required  if  the  pipe  contains  two  rectangular 
knees?  Ans.  1.84918  ft. 

17.  Determine  the  delivery  of  a  2- in.  pipe,  48  ft.  long,  under  a  5-ft. 
head.  Ans.  .1349  cub.  ft.  per  sec. 

What  will  be  the  delivery  if  the  pipe  has  five  small  curves  of  90°  cur- 
vature, the  ratio  of  the  radius  of  the  pipe  to  that  of  the  curves  being 
1:2?  Ans.  .1327  cub.  ft.  per  sec. 

1 8.  The  curved  buckets  of  a  turbine  form  channels  12  in.  long,  2  in. 
wide,  and  2  in.  deep;  the  mean  radius  of  curvature  of  the  axis  is  8  in. 
the  water  flows  along  the  channel  with  a  velocity  of  50  ft.  per  minute. 
What  is  the  head  lost  through  curvature  ?  Ans.  .00138  ft. 

19.  Find  the  maximum  power  transmitted  by  water  in  a  36-inch  pipe, 
the  metal  being  \\  inches  thick  and  the  allowable  stress  2800  Ibs.  per 
square  inch.     If  the  pipe  is  \\  miles  in  length,  find  the  loss  of  power. 

Ans.  576  H.  P.  ;  720.2  ft. -Ibs. 

20.  Find  the  diameter  of  a  pipe  \  mile  long  to  deliver  1500  gallons  of 
water  per  minute  with  a  loss  of  20  feet  of  head.     (/  =  .005.) 

Ans  .1.0135  ft- 

21.  Water   is   to   be   raised    20   ft.    through   a   3O<ft.  pipe    of    6  in. 
diameter.     Find  the   velocity   of   flow,   assuming  that    10  per  cent  of 
additional  power  is  required  to  overcome  friction. 

Ans.  8.44  ft.  per  sec. 

22.  In  a  pipe  3280  ft.  in  length  the  loss  of  head  in  friction  is  83  ft. 
Taking/  —  .0064,  find  the  diameter.  Ans.  1.527  ft. 

23.  A  pipe  2000  ft.  long  and  2  ft.  in  diameter  discharges  at  the  rate 
of  1 6  ft.  per  second.     Find  the  increase  in  the  discharge  if  for  the  last 


1  24  H  V  D  RA  ULICS. 

1000  ft.  a  second  pipe  of  same  size  be  laid  by  the  side  of  the  first  and 
connected  with  it  so  that  the  water  may  flow  equally  well  along  either 
pipe.  Ans.  7.24  cub.  ft.  per  sec. 

24.  A  pipe  of  length  /  and  radius  r  gives  a  discharge  Q.  How  will 
the  discharge  be  affected  (i)  by  doubling  the  radius  for  the  whole 
length  ;  (2)  by  doubling  the  radius  f  :>r  half  the  length  ;  (3)  by  dividing  it 

into  three  sections  of  equal  length,  of  which  the  radii  are  r,  —,  and  —  , 
.respectively  ?  (f  =  coefficient  of  friction.) 

Ans.  i.  New  discharge  = 


r  +   64/A1 


-4 


9'  +  4// 


4228//y  • 

25.  A  24-inch  pipe  2000  ft.  long  gives  a  discharge  of  Q  cubic  feet  of 
water  per  minute.     Determine  the  change  in  Q  by  the  substitution  for 
the  foregoing  of  either  of  the  following  systems  :   (i)  two  lengths,  each 
of  looo  ft.,  whose  diameters  are  24  in.  and  48  in.  respectively;  (2)  four 
lengths,  each  of  500  ft.,  whose  diameters  are  24  in.,  18  in.,  16  in.,  and 
24  in. 

Draw  the  "  plane  of  charge  "  in  each  case. 

Ans.  (i)   Discharge  is  increased   33.2  per  cent  taking  loss  at 

change  of  section  into  account; 
Discharge  is  increased  35.7  per  cent  disregarding  loss 

at  change  of  section. 

(2)  Discharge   is   diminished  45   per   cent    disregarding 
losses  at  change  of  section. 

26.  Q  is  the  discharge  from  a  pipe  of  length  /  and  radius  r  \  examine 
the  effect  upon  Q  of  increasing  r  to  nr  for  a  length  ml  of  the  pipe. 

* 


Ans.  New  discharge  =  Q 


(n*  -  i)2 


«* 

27.  A  reducer,  I  ft.  in  length,  discharges  at  the  rate  of  400  gallons  per 
minute,  and  its  diameter  diminishes  from  12  in.  to  6  in.;  find  the  total 
loss  of  head  due  to  friction.  Ans.  .0055297. 

28.  A  reservoir  of  10,000  square  feet  superficial  area  and   100  feet      / 
deep  discharges  through  a  pipe  24  in.  in  diameter  and  2000  feet  long.    * 
Find  the  velocity  of  flow  in  the  pipe. 

What  should  be  the  diameter  of  the  pipe  in  order  that  the  reservoir 
might  be  emptied  in  two  hours  ?  Ans.   15.36  ft.  per  sec.;  3.67  ft. 

29.  Eight  cubic  feet  of  ore  is  to  be  raised  at  the  rate  of  900  ft.  per 


•.     FLOW   OF   WATER   IN  PIPES.  12$ 

minute  by  a  water-pressure  engine  with  four  single  acting  cylinders  of 
6  in.  diameter  and  18  in.  stroke,  making  60  revolutions  per  minute. 
Find  the  diameters  of  a  supply- pipe  230  ft.  long  for  a  head  of  230  ft., 
disregarding  friction  of  machinery,  etc.  Ans.  4  in. 

30.  A  2-inch  pipe  A  suddenly  enlarges  to  a  3-inch  pipe  B,  the  quan- 
tity of  water  flowing  through  being  100  gallons  per  minute.     Find  the 
loss  of  head  and  the  difference  of  pressure  in  the  pipes     (i)  when  the 
flow  is  from  A  to  B ;  (2)  when  the  flow  is  from  B  to  A. 

Ans.  (i)  Loss  of  head  =    8.639  in- 

Gain  of  pressure-head  =  13.83     " 

(2)  Loss  of  head  =    7.428   " 

Diminution  of  pressure-head  =  29.88     " 

31.  A  3-inch  horizontal  pipe  rapidly  contracts  to  a  i-inch  mouih- 
piece,  whence   the  water  emerges   into   the   air,   the  discharge   being- 
660  Ibs.  per  minute.     Find  the  pressure  in  the  3-inch  main. 

If  the  3-inch  pipe  is  200  ft.  in  length  and  receives  water  from  an 
open  tank,  find  the  height  of  the  tank. 

Ans.   1003.5  Ibs.  Per  sq.  ft.;  19.92  ft. 

32.  The  efficiency  of  an  engine  is  f ;  it  burns  8  Ibs.  of  coal  per  hour 
per  H.P.,  and  works  8  hours  a  day  for  300  days  in  the  year;  the  cost  of 
the  engine  is  $12.00  per  H.P.,  and  the  cost  of  the  coal  is  $3.00  per  ton ; 
4500  gallons  of  water  per  minute  have  to  be  raised  a  height  of  200  ft. 
through   a  pipe  of  which  the  diameter  is  to  be  a  minimum.     Cost  of 
piping  =  $£>  per  lineal  foot,  D  being  the  diameter.    Find  the  value  of  D. 

Ans.  2.923  ft. 

33.  A  reservoir  is  to  be  supplied  with  water  at  the  rate  of  11,000 
gallons    per    minute,  through   a   vertical    pipe   30   ft.    high;    find    the 
minimum  diameter  of  pipe  consistent  with  economy.     Cost  of  pipe  per 
foot  =  &/,  d  being  the  diameter;  cost  of  pumping  =  i  cent  per  H.P. 
per  hour;  original  cost  of  engine  per  H.P.  =  $100.00;  add  10  per  cent 
for  depreciation.     Engine  works  12  hours  per  day  for  300  days  in  the 
year.  Ans.  4.375  ft. 

34.  A    horizontal    pipe  4  in.   in   diameter  suddenly   enlarges  to   a 
diameter   of  6   in.;    find   the   force   required   to  cause   a  flow   of  300 
gallons  of  water  per  minute  through  the  sudden  enlargement. 

Ans.  .06  H.P. 

35.  1000  gallons  per  minute  is  to  be  forced  through  a  system  of 
pipes  AB,  BC,  CD,  of  which  the  lengths  are  100  ft.,  50  ft.,  120  ft.,  and 
the   radii   4  in.,  6   in.,  and   3  in.,    respectively.      Draw  the   plane   of 
charge. 

Ans.  Loss  in  friction  from  A  to  B  =  111.96   ft.;  loss  at  B  —    4.499  ft.; 
"     "        "          "      B  to  C  —      7.372  "       "     "  C  —  14.56     " 
"     "        "          "      C  to  D  =  566.17     " 


126  HYDRA  ULICS. 

36.  A  pipe    4  in.    in  diameter   suddenly  contracts    to    one  3  in.  in 
diameter;    find  the  power  necessary  to  force  250  gallons  per  minute 
through  the  sudden  contraction.  Ans.  1.23997  H.P. 

37.  If  a  pipe  whose  diameter  is  8  in.  suddenly  enlarges  to  one  whose 
diameter  is  12  ins.,  find  the  power  required  to  force  1000  gallons  per 
minute  through  the  enlargement,  and  draw  to  scale  the  plane  of  charge. 

Ans.  Energy  expended  =  .1377  H.P. 

38.  1000  gallons  per  minute  are  forced   through  a  system  of  pipes 
AB,  BC,  CD,  of  which  the  lengths  are  100  ft.,  50  ft.,  and  120  ft.,  and  the 
radii  6  in.,   3  in.,  and  4  in.,  respectively.     Draw  to  scale  the  plane  of 
charge. 

Ans.  Loss  in  friction  from  A  to  B  =    14.744  ft.;  loss  at  B  =  14.56  ft. 

"     "      "         "    B  to  c  =  235.9    " ;   "    "  c=  8.819" 

"      "        "  "     CtoZ>=  134.36    " 

39.  Water  flows  from  a  3-inch  pipe  through  a   i^-inch   orifice   in 
a  diaphragm  into  a  2-inch  pipe.     What  head  is  required  if  the  delivery 
is  to  be  8  cubic  feet  of  water  per  minute  ?  Ans.  2.826  ft. 

40.  500  gallons  of  waiter  per  minute  are  forced  through  a  continuous 
line  of  pipes  AB,  BC,  CD,  of  which  the  radii  are  3  in.,  4  in.,  2  in.,  and 
the  lengths  100  ft.,  150  ft.,  and  80  ft.,  respectively.     Find  the  total  loss 
of  head  (a)  due  to  the  sudden  changes  of  form  at  B  and  C,  (b)  due  to 
friction.     Find  (c)  the  diameter  of  an  equivalent  uniform  pipe  of  the 
same  total  length. 

Ans.  (a)  .1378   ft.;    1.152    ft. 

(b)  3.688  ft.  in  AB;  1.313  ft.  in  BC\  22.393  ft- in  CD. 

(c)  .4212  ft. 

41.  AB,  BC,  CD  is  a  system  of  three  pipes  of  which  the  lengths  are 
looo  ft.,  50  ft.,  and  800  ft.,  and  the  diameters  24  in.,  12  in.,  and  24  in., 
respectively;  the  water  flows  from  CD  through  a    i-inch  orifice   in   a 
thin  diaphragm,   and  the  velocity  of  flow  in  AB  is  2  ft.  per  second. 
Draw  the   plane    of   charge    and    find    the   mechanical    effect  of   the 
efflux. 

Ans.  Loss  at  B  =  -&  ft.;  at  C  =  -/fa  ft.;  in  friction  from  A  to 
B  =  .8  ft. ;  from  B  to  C  =  1.28  ft.;  from  C  to  D  =  .64  ft. ; 
energy  of  jet  =  14,81  if  H.P. 

42.  looo  gallons  per  minute  flows  through  a  sudden  contraction  from 
12  inches  to  8  inches  at  A,  then  through  a  sudden  enlargement  from  8 
inches  to  12  inches  at  B,  the  intermediate  pipe  AB  being   100  ft.  long. 
Draw  the  plane  of  charge. 

Ans.  Loss  at  A  =  .288  ft. ;  at  B  =  .281  ft. ;  in  friction  from  A 
to  B  =  3.499  ft. 

43.  Water  flows  from  one  tube  into  another  of  twice  the  diameter; 
the  velocity  in  the  latter  is  10  ft.     Find  the  head  corresponding  to  the 
resistance.  Ans.  14.0625  ft. 


FLOW  OF   WATER   IN  PIPES.  12 7 

44.  In  a  given  length  /  of  a  circular  pipe  whose  inner  radius  is  r 
and  thickness  <?,  a  column  of  water  flowing  with  a  velocity  v  is  sud- 
denly checked  by  the  shutting  off  of  cocks,  etc.  Show  that 

\e 


in  which  ^  =  head  due  to  the  velocity  vt  E  =  coefficient  of  elasticity, 
E\  =  coefficient  of  compressibility  of  water,  A  =  extension  of  pipe  cir- 
cumference due  to  E. 

45.  A  loo-gallon  tank,  100  feet  above  the  ground,  is  filled  by  a  i^-in. 
pipe  connected  with  an  accumulator  consisting  of  a  3-ft.  cylinder  with  a 
piston   loaded   with   50  tons.     How  long  will  it  take  to  fill  the  tank, 
assuming  that  frictional  resistances  absorb  nine  tenths  of  the  head  and 
that  the  mean  height  of  the  piston  above  the  ground  is  10  feet? 

Ans.  13.9  sees. 

46.  Determine  the  discharge  from  a  pipe  of  12  in.  radius  and  3280  fty 
in  length  which  connects  two  reservoirs  having  a  difference  of  level  of 
128  ft.     Take  into  account  resistance  at  entrance.     Draw  the  plane  of 
charge.  Ans.  48.571  cub.  ft.  per  sec. 

47.  Determine  the  diameter  of  a  clean  iron  pipe  5000  ft.  in  length 
which  connects  two  reservoirs  having  a  total  head  of  40  ft.  and  dis- 
charges into  the  lower  at  the  rate  of  20  cub.  ft.  per  second.     Draw  to 
scale  the  line  of  charge.  Ans.  1.9219  ft. 

48.  The  difference  of  level  between  the  two  reservoirs  is  100  ft.,  and 
they  are  connected  by  a  pipe  10,000  ft.  long.     Find  the  diameter  of  the 
pipe  so  as  to  give  a  discharge  of  2000  cubic  feet  per  minute  (a)  by 
Darcy's  formula,  (b}  assuming  /  =  .0064.     (Allow  for  loss  of  head  at 
entrance.)  Ans.   (a)  2.266  ft.;  (b)  2.360  ft. 

49.  Two  reservoirs  are  connected  by  a  1 2-inch  pipe  ij  miles  long. 
For  the  first  500  yards  it  has  a  slope  of  i  in  30,  for  the  next  half  mile  a 
slope  of  i  in  100,  and  for  the  remainder  of  its  length  it  is  level.     The 
head  of  water  over  the  inlet  is  55  ft.  and  that  over  the  outlet  is  15  ft. 
Determine  the  discharge  in  gallons  per  minute.     (Take/  =  .0064.) 

Ans.  1950.66. 

50.  Two  reservoirs  are  connected  by  a  6-inch  pipe  in  three  sections, 
each  section  being  three  quarters  of  a  mile  in  length.     The  head  over 
the  inlet  is  20  ft.,  that  over  the  outlet  9  ft.     The  virtual  slope  of  the  first 
section  is  i  in  50,  of  the  second  i  in  100,  and  the  third  section  is  level. 
Find  the  velocity  of  flow,  and  the  delivery. 

Ans.  4.5  ft.  per  sec. ;  332  gallons  per  minute. 

51.  A  pipe  5  miles  long,  of  uniform  diameter  equal  to  12  in.,  conveys 
water  from  a  reservoir  in  which  the  water  stands  at  a  height  of  300  ft. 
above  Trinity  high-water  mark,  to  a  reservoir  in  which  the  water  stands 


128  HYDRA  ULICSl 

at  a  height  of  1 50  ft.  above  the  same  datum.  To  what  height  will  water 
rise  in  a  supply-pipe  taken  one  mile  from  the  lower  end?  For  what 
pressure  would  you  design  the  main  at  this  point,  if  it  lies  20  ft.  above 
the  level  of  the  lower  reservoir  ?  Ans.  179.93  ft.;  19  Ibs.  per  sq.  in. 

52.  The  water  surface  in  one  reservoir  is  500  ft.  above  datum,  and  is 
100  ft.  above  the  surface  of  the  water  in  a  second  reservoir  20,000  ft. 
away,  and  connected  with  the  first  by  an  i8-in.  main.     Find  the  delivery 
per  second,  taking  into  account  the  loss  of  head  at  the  upper  entrance. 

53.  Water  surface  of  a  reservoir  is  300  ft.  above  datum,  and  a  4- in. 
pipe  600  ft.  long  leads  from  reservoir  to  a  point  200  ft.  above  datum. 
Find  the  height  to  which  the  water  would  rise  (a)  if  end  of  pipe  is  open 
to  atmosphere,  (b)  if  it  terminates  in  a  i-inch  nozzle.     In  latter  case  find 
longitudinal  force  on  nozzle.     Ans.  (a)  2f  ft.;  (b}  87.52  ft.;  59.693  Ibs. 

54.  The  surface  of  the  water  in  a  tank  is  388  ft.  above  datum  and  is 
connected  by  a  4-in.  pipe  200  ft.  long   with  a  turbine   146  ft.  above 
datum.     Determine  the  velocity  of  the  water  in  the  pipe  at  which  the 
power  obtained  from  the  turbine  will  be  a  maximum.     Assuming  the 
efficiency  of  the  turbine  to  be  85  per  cent,  determine  the  power. 

Ans.  19.928  ft.  per  sec.;  31.895  H.  P. 

55.  A  pipe  12  in.  in  diameter  and  900  ft.  long  is  used  as  an  inverted 
siphon  to  cross  a  valley.     Water  is  led  to  it  and  away  from  it  by  an 
aqueduct  of  rectangular  section  3  ft.  broad  and  running  full  to  a  depth 
of  2  ft.  with  an  inclination  of  i  in  1000.     What  should  be  the  difference 
of  level  between  the  end  of  one  aqueduct  and  the  beginning  of  the 
other  ?  Ans.  575.8  ft. 

56.  Water  flows  through  a  pipe  20  ft.  long  with  a  velocity  of  10  ft. 
per  second.     If  the  flow  is  stopped  in  -^  sec.  and  if  retardation  during 
the  stoppage  is  uniform,  find  the  increase  in  the  pressure  produced. 
(g  =  32  and  the  density  of  the  water  =  62.5  Ibs.  per  cub.  ft.) 

Ans.  62$  cu.  ft.  of  water. 

57.  An  hydraulic  motor  is  driven  by  means  of  an  accumulator  giving 
750  Ibs.  per  square  inch.     The  supply-pipe  is  900  ft.  long  and  4  in.  in 
diameter.     Find  the  maximum  power  attainable,  and  velocity  in  pipe. 
(/=  .0075.)  Ans.  242.4  H.  P.;  21.203  ft-  Per  sec- 

58.  A  2-inch  hose  conveys  2  gallons  of  water  per  second.     Find  the 
longitudinal  tension  in  the  hose.  Ans.  9.18  Ibs. 

59.  Find  the  pumping  H.  P.  to  deliver  i  cub.  ft.  of  water  per  second 
through  a  i-inch  nozzle  at  end  of  a  3-inch  hose  200  ft.  long,/ being  .016, 

Ans.  97.335  H.  P. 

60.  A  volume  of  water  50  ft.  in  length  flowing  through  a  pipe  with  a 
velocity  of  24  ft.  per  second  is  quickly  and  uniformly  stopped  in  one 
tenth  ol  a  second  by  closing  a  stop-valve.     Find  the  increase  of  pressure 
per  square  inch  in  the  pipe  near  the  valve.  Ans.   162.5  Ibs. 

61.  The  surface  of  the  water  in  a  tank  is  286  ft.  above  datum.     The 


I/ 


FLOW  OF   WATER  IN  PIPES.  12$ 

tank  is  connected  by  a  4-in.  pipe  500  ft.  long  with  a  36-in.  cylinder 
170  ft.  above  datum.  Find  (a)  the  velocity  of  flow  in  the  pipe  for  which 
the  available  power  will  be  a  maximum  ;  (b)  the  power.  If  the  piston 
moves  at  the  rate  of  i  ft.  per  minute,  find  (c}  the  pressure  on  the  piston. 
Also  find  the  height  to  which  the  water  would  rise  if  (d)  the  cylinder 
/end  of  the  pipe  were  open  to  the  atmosphere  and  if  (e)  the  pipe  termi- 
nated in  a  nozzle  I  inch  in  diameter,  neglecting  the  frictional  resistance 
of  the  nozzle.  Finally,  find  (/)  the  power  required  to  hold  the  nozzle. 
(Coeff.  of  friction  =  .005.) 

Ans.  (a)  8.93  ft.  per  sec. ;  (ff)  6.85  H.  P.;  (c)  22.8  tons  per  sq.  ft.; 
(d)  3.74  ft.;  (e)  103.8  ft.;  (/)  70,8  Ibs. 

62.  The  conduit-pipe  for  a  fountain  is  250  ft.  long  and  2  in.  in  diam- 
eter ;  the  coefficient  of  resistance  for  the  mouthpiece  is  .32  ;  the  entrance 
orifice  is  sufficiently  rounded,  and  the  bends  have  sufficiently  long  radii 
of  curvature  to  allow  of  our  neglecting  the  corresponding  coefficient  of 
resistance.     How  high  will  a  ^-in.  jet  rise  uuder  a  head  of  30  ft.  ? 

Ans.  1 9. 14  ft. 

63.  The  difference  in  level  of  two  reservoirs  is  250  ft.  and  they  are 
connected  by  a  24-inch  pipe  AB,  6000  ft.  long.     If  f=  .0064,  draw  the 
plane  of  charge.     A  third  reservoir  is  so  placed  that  the  difference 
between  its  level  and  that  of  the  first  (or  highest)  is  100  ft.,  and  is  con- 
nected to  the  main  at  a  point  O  by  a  branch  OC,  3000  ft.  long  and  12  in. 
in  diameter.     Examine  the  distribution. 

Ans.  Upper  reservoir  will  supply  the  two  lower  reservoirs   if 

AO  <  %BO. 

The  two  upper  reservoirs  will  both  discharge  into  the 
,  lower  reservoir  if  AO  >  %BO. 
(x  If  AO  =  2000  ft'.,  the  pressure-head  at  O  =161  ft.;  2/1  =  14.9 

ft. ;  2/2  =  3.02  ft.;  v*  =  14.18  ft. 

If  AO=4ooo  ft.,  the  pressure-head  at  0=96  ft.;  2/1=13.8  ft.; 
2/2  =  6.7  ft.;  2/3  =  15.4  ft. 

64.  A  pipe  24  in.  in  diameter  and  2000  ft.  long  leads  from  a  reservoir 
in  which  the  level  of  the  water  is  400  ft.  above  .datum  to  a  point  B,  at 
which  it  divides  into  two  branches,  viz.,  a  12-in.  pipe  J3C,  1000  ft.  long, 
leading  to  a  reservoir  in  which  the  surface  of  the  water  is  250  ft.  above 
datum,  and  a  branch  BD,  1500  ft.  long,  leading  to  a  reservoir  in  which 
the  surface  of  the  water  is  50  ft.  above  datum.     Determine  the  diameter 
of  BD  when  the  free  surface-level  at  B  is  (a)  300  ft.,  (b)  250  ft.,  and  (c) 
200  above  datum.  Ans.  (a)  1.454  ft.;  (&)  1.783  ft.;  (c)  2.096  ft. 

65.  Two  reservoirs  A  and  B  are  connected  by  a  line  of  piping  MON, 
2000  ft.  in  length.     From  the  middle  point  O  of  this  pipe  a  branch  OP, 
looo  feet  in  length,  leads  to  a  reservoir  C.     The  reservoirs  A  and  (Tare 
200  feet  and  100  feet,  respectively,  above  the  level  of  C.     The  deliveries 
in  MO,  OP,  ON,  in  cubic  feet  per  second,  are  ^-it,  ^-TT,  and  TC,  respec- 


1 30  HYDRA  ULICS. 

lively.     Find  (a)  the  velocities  of  flow  in  MO,  OP,  ON\  (b]  the  radii  of 
these  lengths;  (c)  the  height  of  the  free  surface-level  at  O  above  C. 

Ans.  (a)  1 1. 121  ft.  per  sec.  in  MO;  10.158  ft.  per  sec.  in  OP; 
14.145  ft.  per  sec.  in  OAr. 

(b)  .49976^.;  .41831  ft.;  .26588  ft. 

(c)  150.5  ft.,  very  nearly. 

66.  A  main,  1000  ft.  long  and  with  a  fall  of  5  ft.  discharges  into  two 
branches,  the  one  750  ft.  long  with  a  fall  of  3  ft.,  the  other  250  ft.  long 
with  a  fall  of  i  ft.     The  longer  branch  passes  twice  as  much  water  as 
the  other  and  the  total  delivery  is  47^  cu.  ft.  per  minute.     The  velocity 
of  flow  in  the  main  is  i\  ft.  per  second.     Find  the  diameters  of  the  main 
and  branches.  Ans.  .63245  ft.;  .288ft.;  488ft. 

67.  How  far  can  100  H.P.  be  transmitted  by  a  3^  in.  pipe  with  a  loss 
of  head  not  exceeding  25  per  cent  under  an  effective  head  of  750  Ibs.  per 
square  inch  ?  Ans.   5426.3  ft. 

68.  A  city  is  supplied  with  water  by  means  of  an  aqueduct  of  rect- 
angular section,  24  ft.  wide,  running  4  ft.  deep,  and  sloping  i  in  2400. 
One-fourth  of  the  supply  is  pumped  into  a  reservoir  through  a  pipe  3000 
ft.  long,  rising  25  ft.  in  the  first  1500  ft.,  and  75  ft.  in  the  second  1500  ft. 
The  pumping  is  effected  by  an  engine  burning  2|  Ibs.  or  coal  per  H.P. 
per  hour,  and  working  constantly  through  the  year.     A  percentage  is  to 
be  allowed  for  repairs  and  maintenance;  the  cost  of  the  coal  per  ton  of 
20oolbs.  is  $4 ;  the  prime  cost  of  the  engine  is  $100  per  H.P. ;  the  effi- 
ciency of  the  engine  is  f ;  the  coefficient  of  pipe  friction  is  .0064,  the 
cost  of  the  piping  is  $30  per  ton.     Determine  the  most  economical  diam- 
eter of  pipe,  and  the  H.P.  of  the  engine.      Ans.  4.84  ft. ;  456.455  H.P. 


CHAPTER   IV. 
FLOW  OF  WATER   IN   OPEN   CHANNELS. 

i.  Flow  of  Water  in  Open  Channels. — A  transverse  sec- 
tion of  the  water  flowing  in  an  open  channel  may  be  supposed 
to  consist  of  an  infinite  number  of  elementary  areas  represent- 
ing the  sectional  areas  of  fluid  filaments  or  stream-lines.  The 
velocities  of  these  stream-lines  are  very  different  at  different 
points  of  the  same  transverse  section,  and  the  distribution  of 
the  pressure  is  also  of  a  complicated  character.  Generally 
speaking,  the  side  and  bed  of  a  channel  exert  the  greatest 
retarding  influence  on  the  flow,  and  therefore  along  these 
surfaces  are  to  be  found  the  stream-lines  of  minimum  velocity. 
The  stream-lines  of  maximum  velocity  are  those  farthest 
removed  from  retarding  influences.  If  the  stream-line  velo- 
cities for  any  given  section  are  plotted,  a  series  of  equal 
velocity-curves  may  be  obtained.  In  a  channel  of  symmetrical 


FIG.  78. 

section,  the  depth  of  the  stream-line  of  maximum  velocity 
below  the  water-surface  is  less  than  one  fourth  of  the  depth  of 
the  water,  while  the  mean  velocity-curve  cuts  the  central 
vertical  line  at  a  point  below  the  surface  about  three  fourths  of 
the  depth  of  the  water. 

In  the  ordinary  theory  of  flow  in  open  channels,  the 
variation  of  velocity  from  point  to  point  in  a  transverse  section 
is  disregarded,  and  it  is  assumed  that  all  the  stream-lines  are 
sensibly  parallel  and  move  normally  to  the  section  with  a 
common  velocity  equal  to  the  mean  velocity  of  the  stream. 
With  this  assumption,  it  also  necessarily  follows  that  the 

131 


132 


HYDRA  ULICS. 


distribution  o4"  pressure  over  the  section  is  in  accordance  with 
the  hydrostatic  law. 

Again,  it  is  assumed  that  the  laws  of  fluid  friction  already 
enunciated  are  applicable  to  the  flow  of  water  in  open  chan- 
nels. Thus,  the  resistance  to  flow  is  proportional  to  some 
function  of  the  velocity  (F(v)},  to  the  area  (S)  of  the  wetted 
surface,  is  independent  of  the  pressure,  and  may  be  expressed 
by  the  term  S.F(v).  An  obvious  error  in  this  assumption  is 
that  v  is  the  mean  velocity  of  the  stream  and  not  the  velocity 
of  the  stream-lines  along  the  bed  and  sides  of  the  channel.  In 
practice,  however,  the  errors  in  the  formulae  based  upon  these 
imperfect  hypotheses  are  largely  neutralized  by  giving  suitable 
values  to  the  coefficient  of  friction  (/). 

When  a  constant  volume  (Q)  of  water  feeds  a  channel  of 
given  form,  the  water  assumes  a  definite  depth.  A  permanent 
regime  is  said  to  be  established  and  the  flow  is  steady.  If  the 
transverse  sectional  area  (A)  is  also  constant,  then,  since 
Q  =  vA,  the  velocity  v  is  constant  from  section  to  section  and 
the  flow  is  said  to  be  uniform.  Usually  the  sectional  area  A  is 
variable  and  therefore  the  velocity  v  also  varies,  so  that  the 
motion  is  steady  with  a  varying  velocity.  Any  convenient 
short  stretch  of  a  channel,  free  from  obstructions,  may  be 
selected,  and  treated  without  error  of  practical  importance,  as 
being  of  a  uniform  sectional  area  equal  to  that  of  the  mean 
section  for  the  whole  length  under  consideration. 

2.  Steady  Flow  in  Channels  of  Constant  Section  (A). — 
The  flow  is  evidently  uniform  ;  and  since  A  is  constant,  the 

depth  of  the  water  is  also  con- 
stant, so  that  the  water-surface 
is  parallel  to  the  channel-bed. 
JH  Consider  a  portion  of  the 
stream,  of  length  /,  between  the 
two  transverse  sections  aa,  bb. 

Let  i  be  the  inclination  of 
the  bed  (or  water-surface)  to 
the  horizon. 


Iff 


FIG.  79. 


Let  Pbe  the  length  of  the  wetted  perimeter  of  a  cross-section. 


FLOW  OF   WATER  IN  OPEN   CHANNELS.  133 

Then,  since  the  motion  is  uniform,  the  external  forces 
acting  upon  the  mass  between  aa  and  bb  in  the  direction  of 
motion  must  be  in  equilibrium. 

These  forces  are  : 

(1)  The  component  of  the  weight  of  the  mass,  viz., 

wAl  sin  i  =  wAli  =  wAl—  =  wAk, 

h  being  the  fall  of  level  in  the  length  /. 

Note.  —  When  i  is  small,  as  is  usually  the  case  in  streams, 

-j  =  tan  /  =  sin  /  =  /,  approximately. 

(2)  The   pressures  upon  the  areas  aa  and  bb,  which  evi- 
dently neutralize  each  other. 

(3)  The   frictional  resistance  developed  by  the  sides   and 
bed,  viz., 


Hence 

wAh  -  PlF(v)  =  o, 
or 

FM      Ah 

-^  =  w=m*> 

m  being  the  hydraulic  mean  depth. 

It  now  remains  to  determine  the  form  of  the  function  F(v). 
In  ordinary  English  practice  it  is  usual  to  take 


W  2g 

f  being  the  coefficient  of  friction.     Then 


or 


jig      

v  =  \i  ~~f  y  mi  =  cy  mi. 


1 34  H  YDRA  UL1CS. 

c  being  a  coefficient  whose  value  depends  upon  the  roughness 
of  the  channel  surface  and  upon  the  form  of*its  transverse 
section. 

Prony  and  Eytelwein  adopted  the  formula 

F(v] 

-±—  =  av  -j-  bv*  =  mi, 
w 

and  carried  out  different  experiments  to  determine  the  values 
of  a  and  b. 

According  to  Prony,          -  =  22472.5        and    -r  —  10607.02, 

"  "Eytelwein,   -  =  41688.02        "      -=    8975.43. 

For  a  velocity  of  about  70  ft.  per  minute  Prony's  and 
Eytelwein's  results  give  the  same  value  for  mi.  For  other 
velocities,  Prony's  values  of  mi  are  greater  or  less  than  those 
of  Eytelwein,  according  as  the  velocity  v  is  greater  or  less 
than  70  ft.  If  v,  however,  does  not  differ  very  widely  from 
70  ft.,  the  change  of  value  is  small  and  of  no  practical 
importance. 

For  values  of  v  exceeding  20  ft.  per  minute  the  term  av 
may  be  disregarded  without  practical  error,  and  the  formula 
then  becomes 

mi  =  bv*t 
or 


Hence 

v  =  105  \/mif  according  to  Prony, 

and 

v  =    95  ^/mi,  according  to  Eytelwein, 

giving  as  a  mean 

v  =  loo^mz,  which  is  Beardmore's  formula. 
The  total  head  H  in  a  stream  is  made  up  of  two  parts,  the 


FLOW  OF   WATER   IN  OPEN   CHANNELS.  13$ 

one  required  to  produce  the  velocity  of  flow,  and  the  other 
absorbed  by  the  frictional  resistance.     Thus, 


2g       ?;/     w 

In  long  canals,  and  in  rivers  with  slopes  not  exceeding  3  ft. 

v* 
per  mile,  the  term  —  is  very  small  as  compared  with  the  term 

/  Mv) 

—  ,  and  may  be  disregarded  without  sensible  error. 
m     w 

Note.  —  The  retarding  effect  of  the  air  upon  the  free  surface 
of  a  stream  or  river  has  yet  to  be  determined  by  careful 
observation  and  experiment.  It  may,  however,  be  assumed 
that  the  resistance  offered  by  calm  air  per  unit  of  free  surface 
is  approximately  one  tenth  of  the  resistance  offered  by  similar 
units  at  the  bottom  and  sides  of  smooth  channels.  Thus,  in 

smooth  channels,  if  X  is  the  width  of  the  free  surface,  the 

Y 
wetted  perimeter  is  more  correctly  P  -\- 

In  general,  the  wetted  perimeter  may  be  expressed  in  the 
form  P  -f-  -ip  ft  being  10  for  smooth  channels  and  greater 

than  10  for  rough  channels.  The  value  of  ft  is  obviously 
diminished  by  opposing  winds  and  increased  by  following 
winds. 

3.  On  the  Form  of  a  Channel.—  In  the  formula 

F(v) 

mt'=     *T' 

=  —  J     and     ilss  -yl   are  similarly  related  in  the   deter- 

mination of  v,  the  mean  velocity  of  flow.  If  v  is  constant,  the 
product  mi  must  also  be  constant,  so  that  if  m  increases  i  must 
diminish,  and  vice  versa.  Thus,  in  a  very  flat  country  the  flow 
may  be  maintained  by  making  m  sufficiently  large,  while  again 
if  the  channel-bed  is  steep  m  is  small. 


136  HYDRAULICS. 

The  erosion  caused  by  a  watercourse  increases  with  the 
rapidity  of  flow.  At  the  same  time  the  sectional  area  (A) 
of  the  waterway  also  increases,  so  that  the  velocity  of  flow  v 
diminishes.  Thus  there  is  a  tendency  to  approximate  to  a 
"  permanent  regime  "  when  the  resistance  to  erosion  balances 
the  tendency  to  scour. 

Hence,  throughout  any  long  stretch  of  a  river,  passing 
through  a  specific  soil,  the  mean  velocity  of  flow  will  be  very 
nearly  constant  if  the  amount  of  flow  (Q)  does  not  vary.  Gen- 
erally speaking,  the  volume  conveyed  by  a  river  increases  from 
source  to  mouth  on  account  of  the  additions  received  from 
tributaries,  etc.  Since  Q  increases,  A  must  also  increase  ;  and 
if  mi  or  v  is  to  remain  constant,  i  must  diminish.  It  is  also 
observed  that  the  surface  slopes  of  large  rivers  diminish  gradu- 
ally from  source  to  mouth. 

Again,  various  problems  relating  to  the  proper  sectional 
form  of  a  channel  may  be  discussed  by  means  of  the  formulae 


'A  . 

v  =  c  \mi  = 


and 


Suppose  the  slope  to  be  constant.     Then 

A 

v*  is  proportional  to  75 

and 

A9 
Q*  is  proportional  to  —p.  ' 

PROBLEM  I.  The  section  of  the  waterway  being  a  rectangle 
of  width  x  and  depth  y,  and  of  given  area  (A  —  xy\  it  is 
required  to  find  the  ratio  of  x  to  y  for  which  the  velocity  of 
flow  (v)  will  be  a  maximum.  Then  dv  =  o,  and  therefore 

P.dA—A.dP 

P*  ' 


FLO W  OF   WATER  IN  OPEN   CHANNELS. 


137 


Hence 

PdA-AdP=o. 
But  dA  —  o  =  xdy  -\-ydx,  and 
therefore  also  ^ 

dP  —  o  =  dx  +  2dy,  J 

since 

P=x  +  2y. 
Hence, 


FIG.  80. 


and  the  mean  hydraulic  depth 

_  A  _       xy       _y 
~P~x-\-2y~2 

=  one  half  of  the  depth  of  the  water. 

The  same  results  follow  if   the  discharge  Q  instead  of  v 
is  to  be  a  maximum.     In  such  case 


dQ 


M'\ 
-  o  -  d\-p)  = 


.dA  -  A*.dP 


and  therefore  $PdA  —  AdP  —  o. 

But  dA  =  o,  and  therefore  dP  =  o.     Hence,  etc. 

Note. — The  same  results  also  follow  if,  instead  of  A  being 
given,  the  wetted  perimeter  P  is  to  be  a  minimum,  since  then 
dP  =  o,  and  therefore  also  dA  =  o. 

PROBLEM  II.     The  waterway  being  trapezoidal  in  section, 


FIG.  81. 


of  bottom  width  x,  depth  y,  and  sides  sloping  at  a  given  angle 
Q  to  the  horizontal,  it  is  required  to  find  the  ratio  of  x  to  y 
which,  for  a  given  wetted  perimeter  (P^  or  area  (A),  will  make 
the  velocity  of  flow  or  the  discharge  a  maximum. 


HYDRAULICS. 


As  in  Problem  I, 

dA  =  o     and     dP  =  o. 


But  A  =  (x  +y  cot  6)y     and     P  =  x  +  2y  cosec  6. 
Hence 

^£4=0=  ydx  +  <^/O  +  27  cot  0) 
and 

</P  =  o  —  dx  +  */j/  .  2  cosec  0. 

Therefore 

x  -4-  2y  cot  #  dx 

—  !  —  -  ---  =  —  -j-  =  2  cosec  0. 
dy 


y 
Hence 

x  :=  2i/(cosec  0  —  cot  0)  =  2i/ 
and  therefore 


-  cos  0 


^      ==  2v  tan  —  t 
sin  0  2 


x  0 

—  =  2  tan  —  . 

/  2 


Then  mean  hydraulic  depth 
A        (*-\-y  cot  0)y 


j/2  —  cos  6K       y 
0  =  2(2  -  cos  0}  ~  2 


P      x  -|-  27  cosec 
=  one  half  of  the  depth  of  the  water. 

The  section  may  be  easily  sketched  as  in  Figs.  82  and  83. 


G 

FIG.  82. 


From  the  middle  point  C  of  AB,  the  bottom  width,  draw 
CF  at  right  angles  to  AB  and  equal  in  length  to  the  depth  of 
the  water.  Then 

AB  _  0 

-  2  tan     , 


0  being  the  given  slope  of  the  sides. 


FLOW  OF   WATER   IN  OPEN   CHANNELS.  139 

With  F  as  centre  and  FC  as  radius  describe  a  circle.  From 
the  points  A  and  B  draw  tang.ents  to  touch  this  circle  at  D 
and  E.  FA  evidently  bisects  the  angle  CAD.  Therefore 

CAD  CF        CF  6 

tan  -       =  tan  CAP  =  =  -      =  =  cot  2' 


Hence  TT  —  CAD  =  #,  and  ^/?,  /?£  have  the  slope  required. 

PROBLEM  III.  To  find  the  proper  sectional  form  of  a 
channel  of  bottom  width  2a  so  that  the  mean  velocity  of  flow 
may  be  constant  for  all  depths  of  water. 

Let  x,  y,  Fig.  84,  be  the  co-ordinates  of  any  point  P  in  the 
profile  referred  to  the  middle  point  O  of  AB,  the  bottom  width,. 
as  origin  and  let  s  be  the  length  of  A  P. 


FIG.  84. 

Since  v  is  to  be  constant  m  must  also  be  constant,  and 
therefore 

•    A=,[y^_= 

which  may  be  written 

/  ydx  =  m(s  +  a). 
Differentiating, 

ydx  =  mds  —  m^dx* 
and  therefore 

dx  dy 

m  ~~  (/  _  w? 
Integrating, 

x  

m~      *>e  \s 

c  being  a  constant  of  integration. 


1 4°  H  YDRA  ULICS. 

When  x  =  o,  y  =  a,  and  therefore 


o  =  log,  (a  +  *V  -  M2)  +  c  =  loge&  +  c, 
where  b  =  a  -f-  j/#a  —  ^2.     Hence 


£  =  te 


or 


is  the  equation  to  the  required  profile,  which,  as  may  be  easily 
shown,  is  a  curve  which  flattens  very  rapidly. 

PROBLEM  IV.     If  water  flows  through  a  circular  aqueduct, 
find  the  angle  6  subtended  at  the  centre  by  the  wetted  perim- 
eter, for  which  the  velocity  of  flow  is  a 
maximum. 

Let  r  =  radius  of  aqueduct. 


Area  of  waterway  =  —  (6  —  sin  0). 


Wetted  perimeter  =  r6. 
FIG.  85.  Then 


r  0  —  sin  0       r  I         sin 


m  =  — -ft — 

2  C^ 

sin  0 
Now  v  is  to   be*  a   maximum  and  therefore  — -^—  must  be  a 

minimum.     Hence 

0  cos  0  -  sin  0  , 


/sin  0\  0COS 

4—)=°=- 


and  therefore  0  cos  —  sin  0  =  o. 

Hence  6=  tan  0,  and  the  angle  0  in  degrees  is  about  77°  27'. 


FLOW  OF   WATER  IN  OPEN   CHANNELS. 


141 


Also,  the  mean  hydraulic  depth  =  —  ^i  ---  -r—  ) 


=  -  (i  _  cos  d) 


=  rsin—  =  .  39  X  r. 

PROBLEM  V.  A  channel  of  given  slope  has  a  given  surface- 
width  AC,  vertical  sides  AB  (=yl)  and  CD  (=7,)  of  given 
depths,  and  a  curved  bed  BD  (=  L)  of  given  length. 


FIG.  86. 


The  amount  and  velocity  of  flow  in  the  channel  will  be  a 
maximum  when  the  form  of  the  bed  BD  is  a  circular  arc.  This 
can  be  easily  proved  as  follows  : 


Since  the  slope  is  constant,  v  oc 


/~A 

a  \f  -p. 


But  P  (=  L  -\-  y^  +  >0  is  a  constant  quantity,  and  therefore 
v  and  also  Q  will  be  a  maximum  when  ^4  is  a  maximum. 

Hence,  too,  the  area  between  the  chord  BD  and  the  curve 
must  be  a  maximum,  and  therefore  the  curve  must  be  a  circu- 
lar arc.  The  proof  of  this  by  the  Calculus  of  Variations  is  as 
follows  : 

Take  O  in  CA  produced  as  the  origin,  OC  as  the  axis  of  x, 
and  the  vertical  through  O  as  the  axis  of  y.  Then 


ydx  is  to  be  a  maximum. 


1 42  H  YD  RA  ULICS. 

Also, 


dy 
is  a  given  quantity,  OA  being  =  JT,  ,  OC  =  x^ ,  and  Hr 


Let  V  =•  y  -f-  a  Vi  -\-  p\  a  being  some  constant. 
Then 

/*; 

/      F.  dx  is  to  be  a  maximum, 

*X  ^i 


and  therefore 


that  is, 


and  thus 


^  +  ~^=r^  =  ^  • 


Therefore 


^       /     '   Va*  -  (c,  -  <y)*  ' 
Integrating, 


the  equation  to  a  circle  of  radius  a. 

Hence  the  profile  BD  is  a  circular  arc. 

The  maximum  depth  of  the  channel  is  cl  —  a. 

The  constants  c1 ,  c^ ,  a  can  be  found  from  the  three  con- 
ditions that  the  arc  is  of  given  length  and  has  to  pass  through 
the  two  fixed  points  B  and  D. 

4.  Flow  in  Aqueducts. — The  velocity  v  depends  upon  m 

A 
(m  =  -   and  therefore   upon   the  depth   of  the  water  in   the 


FLOW  OF   WATER  IN  OPEN   CHANNELS.  143 

aqueduct.  For  some  definite  depth  the  velocity  will  be  a 
maximum.  If  the  water  fills  the  aqueduct,  the  aqueduct  be- 
comes a  pipe,  and  the  formula  for  channel-flow  ought  to  change 
suddenly  so  as  to  agree  with  that  for  pipe-flow.  The  theory  is 
thus  imperfect. 

5.  River-bends. — The  following  explanation  is  due  to  Pro- 
fessor  James  Thomson  (Inst.  Mechl.  Engs.,  1879  5  Pi'oc.  Royal 
Soc.  1877).  In  rivers  flowing  in  alluvial  plains,  the  curvature 
of  the  windings  which  already  exist  tends  to  increase  owing  to 
the  scouring  away  of  material  from  the  outer  bank  and  to  the 
deposition  of  detritus  along  the  inner  bank.  The  sinuosities 
often  increase  until  a  loop  is  formed,  with  only  a  narrow  isth- 
mus of  land  between  two  encroaching  banks  of  a  river.  Finally 
a  cut-off  occurs,  a  short  passage  for  the  water  is  opened 
through  the  isthmus,  and  the  loop  is  separated  from  the  river- 
course,  taking  the  form  of  a  horseshoe-shaped  lagoon  or  swamp. 
The  ordinary  supposition  that  the  water  always  tends  to  move 
forward  in  a  straight  line,  rushing  against  the  outer  bank  and 
wearing  it  away,  and  at  the  same  time  causing  deposits  at  the 
inner  bank,  is  correct,  but  it  is  very  far  from  being  a  complete 
explanation  of  what  takes  place. 

When  water  flows  round  a  circular  curve  under  the  action 
of  gravity  only,  it  takes  a  motion  like  that  in  a  free  vortex. 
Its  velocity  parallel  to  the  axis  of  the  stream  is  greater  at  the 
inner  than  at  the  outer  side  of  the  curve. 

Thus,  too,  the  water  in  a  river-*bank  flows  more  quickly 
along  courses  adjacent  to  the  inner  bank  of  the  bend  than 


FIG.  87. 

along  courses  adjacent  to  the  outer.  The  water,  in  virtue  of 
centrifugal  force,  presses  outwards  so  that  the  water-surface  of 
a  transverse  section  (Fig.  87)  has  a  slope  rising  upwards  from 


144 


HYDRA  ULICS. 


the  inner  to  the  outer  bank.  Hence  the  free  level  for  any 
particle  of  the  water  near  the  outer  bank  is  higher  than  the 
free  level  for  any  particle  in  the  same  transverse  section  near 
the  inner  bank,  but  the  tendency  to  flow  from  the  higher  to 
the  lower  level  is  counteracted  by  centrifugal  action.  Now 
the  water  immediately  in  contact  with  the  bottom  and  sides 
of  the  course  is  retarded,  and  its  centrifugal  force  is  not  suf- 
ficient to  balance  the  pressure  due  to  the  greater  depth  at  the 
outside  of  the  bend.  This  water  therefore  tends  to  flow  from 


FIG.  88. 


the  outer  bank  towards  the  inner  (Fig.  88),  carrying  with  it 
detritus  which  is  deposited  at  the  inner  bank.  Simultaneously 
with  the  flow  of  water  inwards,  the  mass  of  the  water  must 
necessarily  flow  outwards  to  take  its  place. 

6.  Value  of/. — The  value  of /depends  upon 

(a)  the  roughness  of  the  sides  and  bed ; 

(b]  the  velocity  of  flow  ; 

(c)  the  dimensions  of  the  transverse  section ; 

(d]  the  slope  of  the  channel-bed. 
An  average  mean  value  of /is  .00757. 


FLO W  OF   WATER  IN  OPEN  CHANNELS.  145 

Weisbach  has  proposed  to  take 


the  values  of  a  and  /?,  obtained  as  the  results  of  255  experi- 
ments, being  a  =  .007409,  ft  =  .192,  so  that 

,    .0014225 
/=.  007409+ . 

Darcy  and  Bazin  assume  f  to  be  given  by  an  expression  of 
the  form 

ft 


giving  the  following  values  of  a  and  ft  as  the  results  of  their 
experiments : 

In  very  smooth   channels,  with  sides  of  planed  timber  or 
rendered  in  cement, 

.000316 
a  =  .00316,  /3  =  .1  ;       .'. /=  .00316  +  -         — . 

In  smooth  channels  with  sides  of  planks,  brick-work,  or 
ashlar 


,    .0009223 
a  =  .00401,  ft  =  .23  ;     /.  /=  .00401  +  -  --. 

In  rough  channels  with  sides  of  rubble  masonry  or  pitched 
with  stone 

<*=:. 00507,     0  =  .82;     .-./  =  .00507  +  :  °^574. 
In  very  rough  channels  in  earth 

or  =  .  00592,      0  =  4.i;     .-./=.  00592+  '°21272. 


146  HYDRA  ULICS. 

In  torrential  streams  encumbered  with  detritus 


a  =  .00846,     /?  =  8.2  ;     .-./=  .00846  +  '—. 


Ganguillet  and  Kutter,  taking  the  formula 


v  =  c     m  = 

kave  endeavored  to  obtain  a  more  correct  value  of  c  by  a  care- 
ful investigation  of  : 

(a)  The  experimental  results  of  Darcy  and  Bazin.     These 
results  show  that  the  value  of  c  depends  upon  the  roughness  of 
the  channel  and  also  upon  its  dimensions.     The  values  given 
for  a  and  ft  are  different  for  different  classes  of  channel  even 
when  the  dimensions  are  infinite.     But  while  in  small  channels 
the  influence  of  differences  of  roughness  upon  the  flow  must 
be  very  great,  it  is  certainly  more  than  probable  that  this  in- 
fluence diminishes  as  the  section  of  the  channel  increases,  and 
that  it  will  be  nil  in  the  case  of  an  indefinitely  large  channel. 

(b)  The  measurements  of  Humphreys  and  Abbott  on  the 
Mississippi,  a  stream  of  very  large  section  and  of  very  low 
slope. 

(c)  Their  own  gaugings  in  the  regulated  channels  of  certain 
Swiss  torrents  with  exceptionally  steep  slopes   and    running 
through  extremely  rough  channels. 

(d)  The  effect  of  the  slope. 

From  the  Mississippi  data  it  was  found  that 

c  —  256  for  a  slope  of  .0034  per  looo 
and 

c  =  154     "  "     "       "  .02        "       " 

Thus  c,  and  therefore  also  the  discharge,  will  be  subject 
to  considerable  variations  in  the  case  of  large  streams  with  low 
slopes.  The  value  of  c  does  not  vary  much  with  the  slope  in 


FLOW  OF   WATER   IN  OPEN  CHANNELS.  147 

small  rivers.     Proceeding  in  a  purely  empirical  manner,  Gan- 
guillet  and  Kutter  arrived  at  the  formula 


C  = 


where  n  is  a  coefficient  depending  only  on  the  roughness  of 
the  channel  sides  and  bed,  while  A  and  /  are  new  coefficients 
whose  values  remain  to  be  determined. 

Now  c  depends  upon  the  slope  i  and  decreases  as  i  in- 
creases.    This  may  be  allowed  for  by  taking 


so  that 


c  = 


the  form  finally  adopted  by  Ganguillet  and  Kutter. 

The  values  given  for  the  constants,  the  unit  being  a  foot, 
are 

0  =  41.6;     /=i.8n;    p  •=•  .00281  ;     n  =  .008  to  .05. 

The  following  table  gives  the  values  of  n  which  will  be 
found  of  most  use  in  practice : 

In  a  channel  with  sides  of  well-planed  timber n  =  .009 

"  "  "         "     rendered  with  cement n  =  .01 

In  a  channel  with  sides  rendered  with  a  mixture  of 

3  of  cement  to  I  of  sand n  =  .01 1 

In  a  channel  with  sides  of  unplaned  planks n  =  .012 

"     "   ashlar  or  brickwork n  =  .013 

"  "         "          "         "     "  canvas  on  frames w  =  .oi5 

"     "    rubble  masonry n  =  .017 


1 48  H  YDRA  ULICS. 

In  rivers  and  canals  in  very  firm  gravel n  =  .02 

In  rivers  and  canals  in  perfect  order  and  free  from 

detritus  (stones  and  weeds) n  =  .025 

In  rivers  and  canals  in  moderately  good  order,  not 

quite  free  from  stones  and  weeds n  —  .03 

In  rivers  and  canals  in  bad  order,  with  weeds  and 

detritus   n  =  .035 

In  torrential  streams  encumbered  with  detritus n  =  .05 

To  the  above  Jackson  adds  the  following  classification  for 
artificial  canals : 

In  canals  in  very  firm  gravel  in  perfect  order n  =  .02 

u        u       "    earth  above  the  average  order n  =  .0225 

"        "       "       "      in  fair  order n  =  .025 

"        "       "       "      below  the  average  order #  =  .0275 

In  canals  in  earth  in  rather  bad  order,  partially  over- 
grown with  weeds  and  obstructed  with  detritus,  n  =  .03 

The  difficulty  of  properly  selecting  the  value  of  n  is  due  to 
the  fact  that  there  is  no  absolute  measure  of  the  roughness  of 
channel-beds. 

In  Cunningham's  experiments  on  the  Ganges  c  varied 
from  48  to  130. 

In  Humphreys  and  Abbott's  experiments  on  the  Missis- 
sippi c  varied  from  53  to  167,  the  units  in  each  case  being  a 
foot  and  a  second. 

7.  Variation  of  Velocity  in  different  parts  of  the  trans- 
verse section  of  a  stream. 

Assumptions. — (a)  That  the  stream  is  of  uniform  depth  h 
and  of  indefinite  width. 

(b)  That  the  fluid  filaments  flow  across  the  section  in  sen- 
sibly parallel  lines. 

(c)  That  a  permanent  regime  has  been  established,  and  that 
the  flow  is  uniform.     The  pressure  in  the  section  is  therefore 
distributed  in  accordance  with  the  hydrostatic  law. 

(d)  That  the  resistance  to  the  relative  sliding  of  consecutive 
filaments  is  of  the  nature  of  viscous  resistance. 


FLOW  OF   WATER  IN  OPEN  CHANNELS. 


149 


Let  Fig.  89  represent  a  portion  of  a  vertical  longitudinal 
section  of  the  stream  intersected  by  two  transverse  sections 
AB,  CD,  I  being  the  distance  between  them. 


FIG  89. 

Consider  a  thin  layer  abed  of  thickness  dy  and  width  b, 
bounded  by  the  sections  AB,  CD,  and  by  the  planes  ad,  be,  at 
depths y  and/  -j-  dy,  respectively,  below  the  free  surface. 

The  forces  acting  upon  the  layer  in  the  direction  of  motion 
are : 

(1)  The  pressures  on  the  ends  ab,  cd,  which  evidently  neu- 
tralize each  other. 

(2)  The  component  of  the  weight— wbl.  dy .  sin  i  =  wbli .  dy  ; 
i  being  the  slope  of  the  bed. 

(3)  The  viscous  resistances  on  the  lateral  faces  of  the  layer 
under  consideration.    These  are  nil,  since  in  a  stream  of  indefi- 
nite width  there  will  be  no  relative  sliding  between  abed  and 
the  vertical  faces  on  each  side. 

(4)  The  viscous  resistances  along  the  planes  ad  and  be. 
The   frictional  resistance  to   distortion,   i.e.,   to    shearing, 

along  such  planes  is  found  to  be  proportional  to  the  shear  per 
unit  of  time,  and  is  measured  by  the  shear  per  unit  of  area  at 
the  actual  rate  of  shearing.  The  coefficient  of  viscosity,  or 

shear  per  unit  of  area 
simply  the  viscosity,  is  the  quotient    -. — 

shear  per  unit  of  time 

and  defines  that  quality  of  the  fluid  in  virtue  of  which  it  resists 
a  change  of  shape. 

Adopting  Navier's  hypothesis, 

di} 
the  viscous  resistance  along  ad  =  —  kbl--. 


150 


HYDRAULICS. 


k  being  the  coefficient  of  viscosity.     The  sign  is  negative  as, 

dv  . 
since  v  increases  withjj/,  -y-  is  positive,  and,  at  the  same  time, 

the-  action  of  the  layers  above  ad  is  of  the  character  of  a  re- 
tardation. 


dv 

The  viscous  resistance  along  be  =  kbl-j- 

dy 


Then,  as  the  motion  is  uniform, 


dv 

kbl  .  d-j- 
y 


; 

dy 


wbli .  dy  -  kbl^-  +  kbl~  +  kbl^dy  = 
dy  dy  dy    " 


Hence 


w 


Integrating  twice, 


(I) 


a  and  vs  being  constants  of  integration. 

It  is  evident  that  vs  is  the  surface-velocity,  i.e.,  the  value 
of  v  when  y  —  o. 

The  equation  may  be  written  in  the  form 

ka*  wi  f         ka\* 


FIG.  90. 


dv 


=  °) 


and 


Thus   the  velocity  -  curve  is  a  parabola 

ka 
having  a  horizontal  axis  at  a  depth  Y '=  — r 


wi 


below   the    free   surface.     This   is   also   the 
depth  of  the  filament  of  maximum  velocity 


(3) 


FLOW  OP   WATER   IN  OPEN   CHANNELS.  !$! 

Hence,  by  equations  I  and  3, 

wi 
v  =  v^--k(y-  Y}\      .   ..    ,    .    .     (4) 

Let  vm  be  the  "  mean  "  velocity  for  the  whole  depth  h. 
Let  v\  be  the  mid-depth  velocity.     Then 


f 


and 

with  V 


(6) 

Hence 

witf 


a  result  upon  which  Humphreys  and  Abbott  have  based  a  rapid 
method  of  gauging  rivers. 

Let  vb  be  the  bottom  velocity,  i.e.,  the  value  of  v  when 
y  =  h.     Then  by  equation  4, 


wi 


and  therefore 

^^      f  1  T/"\2  /O\ 

^max  —   Vb  =   — 7-(/2  —    Yy (8) 

2k  ^ 

When  the  filament  of  maximum  velocity  was  below  the  free 
surface  Bazin  found  the  value  of  the  difference  z>max  —  vb  to  be 
constant.  Take 

IllUX  O  sy       7y       \  / 


IS2  HYDRA  ULICS. 

Then  the  general  equation  (4)  of  the  velocity-curve  becomes 

•     .     ....     (9) 


Now  if  Y'—  o,  i.e.,  if  the  filament  of  maximum  velocity  is 
in  the  free  surface, 

H  P=tw-A^. 

But  in  such  case  Bazin's  experiments  led  to  the  relation 


Hence 

^=36.3 

and  the  general  equation  of  the  velocity-curve  becomes 

^iv-  FV 


.....     (10) 


This  is  Bazin's  formula,  and  it  agrees  well  with  his  experi- 
ments on  artificial  channels  and  also  with  the  results  of 
experiments  on  the  Saone,  Seine,  Garonne,  and  Rhine.  It 
was  found  that 

*7) 

-  —  1.17  in  the  Rhine  at  Basle  and  ranged  from  i.i  to  1.13 

^»« 

in  the  others; 


36.3^  i/      . 
(h  _  KY"     y  between  r3  and  2o; 

Y 

—  =  .33  in  some  artificial  channels  and  ranged  from  O  to  0.2 

in  the  other  cases  ; 
*W  —  vb  ranged  from  Jz/max  to  i^max. 

These  results  are  not    in   agreement  with   the  Mississippi 
measurements. 


FLOW  OF   WATER   IN  OPEN  CHANNELS.  153 

Note. — When  the  filament  of  maximum  velocity  is  in  the 
free  surface,  Y  =  o,  and  therefore,  by  equation  5, 

wih* 

«TI  —       191  

um   —   ^max   ~~"     z-  7     > 

and  by  equation  8, 

wit? 


Hence,  combining  these  two  equations, 


Boileau  assumes  that    the  velocity-curve  is  given   by  the 
equation 

..     .....     (12) 


below  the  filament  of  maximum  velocity,  being  MMl  in  Fig.  91, 
and  by  the  equation 

v  =  a-Bf  +  Cy (13) 

above  the  filament  of  maximum  velocity,  being  MM9  in  Fig.  92. 

Let  vs  be  the  surface-velocity,  i.e.,  the 
value  of  v  when  y  —  o.  Then,  by  equa- 
tion 13, 

vs  =  a. 

Also,  the  two  equations  (12)  and  (13) 
must  each  give  the  same  value  for  the 
maximum  velocity  (zw),  and  therefore 

A  -  BY*  =  vmax  =  a  —  BY*  +  CY,  FlG'  9I> 

from  which 

A  —  a      A  —vs 


Again,  taking  A  =  z/max  +  ^  Boileau  deduced  experimen- 
tally that  d  is  sensibly  constant  for  different  streams. 


1  54  HYDRA  ULICS. 

But  A  =  ?w  +  d  =  A  -  B  Y*  +  d,  and  therefore  B 
Hence  Boileau's  equation  becomes 


for  points  below  the  filament  of  maximum  velocity,  and 

V  =  V,  -  *'  +  (ZW  +  d-  ». 


for  points  above  the  filament  of  maximum  velocity. 

8.  Relations  between  Surface,  Mean,  and  Bottom  Ve- 
locities. —  Bazin  deduced  from  his  experiments  on  canals  the 
relation 

,  —  vm 

vm  =  vs  —  25.4  Vmt  =  vs  —  25.4—, 


where  c  —  V  -~.     Therefore 


cvs 

vm  = 


-  c+  25.4 
Darcy  and  Bazin  give  the  relation 


10.87  ^wt  =  vb  +  10.87—. 


Therefore 

v    = 

~ 


C  —    I0.8/ 

A  mean  value  of  c  is  45.7,  which  makes 

vm  =  1.312. zv        ib 

Dubuat   gives   the   following   table   of   maximum  bottom 
velocities  consistent  with  stability : 


FLOW  OF   WATER  IN  OPEN  CHANNELS. 


155 


Nature  of  Canal  Bed.  Vj,. 

Soft  earth 0.25 

Loam 0.50 

Sand i.oo 

Gravel 2.00 

Pebbles 3.40 

Broken  stone,  flint 4.00 

Chalk,  soft  shale 5.00 

Rock  in  beds 6.00 

Hard  rock , 10.00 

TABLE   OF    MAXIMUM  VELOCITIES   FROM    INGENIEURS 
TASCHENBUCH. 

Nature  of  Canal-bed.                                    vs  vm  vb 

Slimy  earth  or  brown  clay 49  .36  .26 

Clay 98  .75  .52 

Firm  sand 1.97  1.51  1.02 

Pebbly  bed 4.00  3.15  2.30 

Boulder  bed 5.00  4.03  3.08 

Conglomerate  of  slaty  fragments 7.28  6.10  4.90 

Stratified  rocks 8.00  7.45  6.00 

Hard  rocks 14.00  12.15  10.36 


TABLE  OF  VISCOSITY  OF  WATER   AND   MERCURY. 
(From  Everett's  System  of  Units.) 


WATER. 


MERCURY. 


Temp. 

(Cent.) 


o 

5 
10 

15 

20 
25 
30 


Viscosity. 


.Ol8l 
.0154 

•0133 
.0116 
.OIO2 
.OOQI 
.0081 


Temp. 
(Cent.) 


35 
40 
45 
50 
60 
80 
90 


Viscosity. 


.0073 
.0067 
.0061 
.0056 
.0047 
.0036 
.0032 


Temp. 
(Cent.) 


Ou 
IO 

18 

99 
154 
197 

249 


Viscosity. 


.0169 
.0162 
.0156 
.0123 
.0109 
.OIO2 
. 00964 


Temp. 

(Cent.) 


315 
340 


Viscosity. 


.00918 
.00897 


156  HYDRA  ULICS. 

The  viscosity  is  given  by 
_.oi83 

and  by 


.0369^ 


,  according  to  Meyer ; 


j| .00131,  according  to  Slotte; 

/  being  the  temperature  centigrade. 

9.  Flow  of  Water  in  Open  Channels  of  Varying  Cross- 
section  and  Slope, 

Assumptions. — (a)  That  the  motion  is  steady. 
Thus  the  mean  velocity  is  constant  for  any  given  cross- 
section,  but  varies  gradually  from  section  to  section. 

(b)  That  the  change  of  cross-section  is  also  gradual. 

(c)  That,  as  in  cases  of  uniform  motion,  the  work  absorbed 
by  the  frictional  resistance  of  the  channel-bed  and  sides  is  the 
only  internal  work  which  need  be  taken  into  consideration. 


Xy 

FIG.  Q2. 


Let  Fig.  92  represent  a  longitudinal  section  of  the  stream. 
The  fluid  molecules  which  are  found  in  any  plane  section  db 
at  the  commencement  of  an  interval  will  be  found  in  a  curved 
surface  dc  at  the  end  of  the  interval,  on  account  of  the  differ- 
ent velocities  of  the  fluid  filaments. 

Suppose  that  the  mass  of  water  bounded  by  the  two  trans- 
verse sections  ab,  ef,  comes  into  the  position  cdhg  in  a  unit  of 
time.  Then  the  change  of  kinetic  energy  in  this  mass  is  equal 
to  the  algebraic  sum  of  the  work  done  by  gravity,  of  the  work 
done  by  pressure,  and  of  the  work  done  against  the  frictional 
resistance. 

Change  of  Kinetic  Energy. — This  is  evidently  the  difference 


FLOW  OF   WATER   IN  OPEN  CHANNELS.  I  57 

between  the  kinetic  energies  of  the  masses  efgh  and  abed, 
since,  as  the  motion  is  steady,  the  kinetic  energy  of  the  mass 
between  cd  and  ef  remains  constant. 

Let  A1  be  the  area  of  the  cross-section  ab. 
"      «j    "     "    mean  velocity  across  this  section. 
*'      v     "     "    velocity  at  this  section  of  any  given  fluid, 

filament  of  sectional  area  a. 
Let  v  =  ul±  V. 
Then 

Aji,  =  2(av)     and     2(aV)  =  O. 

The  kinetic  energy  of  the  mass  abed 


Since  S(aV)  —  o     and     3«x  ±  V  —  2«,  +  v. 

Now  2«j  +  v  is  evidently  positive.     Hence  the  kinetic  en- 
ergy of  the  mass  abed 


a'  being  a  coefficient  of  correction  whose  value  depends  upon 
the  law  of  the  distribution  of  the  velocity  throughout  the  sec- 
tion ab.  It  is  positive  and  greater  than  unity.  Assume  that 
a  has  the  same  value  for  the  sections  ab  and  ef.  Then  if  A^ 


158  H  YDRA  ULICS. 

#a,  are  the  area  and  mean  velocity  at  the  transverse  section  ef, 
the  kinetic  energy  of  the  mass  efgh 

=  <x~A^' 

Hence  the  change  of  kinetic  energy  in  the  mass  under  con- 
sideration 


g  2 

since        A^ut  =  Q  =  A.u^ 


Work  done  by  Gravity.  —  Consider  any  fluid  filament  mn, 
the  depth  of  m  below  the  surface  being  y^  and  of  n,  y^. 
Let  z  be  the  fall  in  the  surface-level  from  a  to  e. 
Then  the  fall  from  m  to  n 


and  the  work  done  by  gravity  on  the  elementary  volume  dQ 
in  a  unit  of  time 


Work  done  by  Pressure. 

The  pressure  per  unit  of  area  at  m  —  wyl  -\-p0  ; 


0  being  the  atmospheric  pressure. 

Hence  the  work  due  to  these  pressures  per  unit  of  time 

=  dQ(wy,  +  A)  -  dQ(wy,  +/.), 

=  w  . 


Thus  the  total  work  done  by  gravity  and  by  pressure 


=  2(w  .dQ.z)  =  wQz, 
for  the  mass  under  consideration. 


FLO  W  OF   WATER   IN  OPEN   CHANNELS.  159 

Work  absorbed  by  Friction. — Consider  a  thin  lamina  of  water 
of  thickness  ds,  bounded  by  the  transverse  planes  xx,  yy,  the 
distance  of  xx  from  ab  being  s. 

Since  the  change  of  velocity  is  gradual,  the  mean  velocity 
from  xx  to  yy  may  be  assumed  to  be  constant. 

Let  u  be  this  mean  velocity. 
"    Pbe  the  wetted  perimeter  at  the  section  xx. 
"   A  be  the  area  of  the  waterway  at  the  section  xx. 

Then  the  work  absorbed  by  friction  per  second  from  xx 
to  yy 

=  P.ds.u.F(u\ 
and  the  total  work  absorbed  between  ab  and  ef 


=  <2 

*. 

/  being  the  distance  between  ab  and  ef.     Hence 


a 


and  therefore  z  =  a""*  ~  "'  +    /  - 


2g  J0Aw 

~  .     F(u)        -u*       ,  A 
Take  —  ^  =  /—  and  —  =  m.     Then 
w  2g         P 


2g 


If  the  two  planes  ab  and  ef  are  indefinitely  near  one  an- 
other (Fig.  93),  the  last  equation  evidently  gives, 

j        a      j        f  u*  j  /  \ 

dz  =  —u  .  du  -4-  -  --  as.      .....     (2) 

g  m2g 


160  HYDRA  ULICS. 

which  is  the  fundamental  differential  equation  of  steady  varied 
motion,  dz  being  the  fall  of  surface  level  in 
the  distance  ds. 

In  the  figure  aa'  is  drawn  parallel  to  the 
bed  and  aa"  is  horizontal.  The  distance 
a"  e  may,  without  sensible  error,  be  assumed 
equal  to  dz. 

Also  a"  a'  =  i  .  aa'  —  i  .  ds,  very  nearly. 

ids  —  a'  a"  =  a'e  +  a"  e  =  dh  +  dz.      .     .     .     (3) 

Substituting  the  value  of  dz  from  this  equation   in  equa- 
tion 2, 

i  .  ds  —  dh  =  -u  .  du  +  —  —  .  ds.  .  (4) 

g 


Also,  since  Au  =  Q,  a  constant, 

^4  .  du  +  »  .  dk4  =  o, 

and  dA  =  x  .  dk,  very  nearly,  if  x  is  the  width  of  the  stream. 
Therefore 

Adu  +  ux  .  dh  =  o, 

and  hence,  by  equation  4, 

.     ,         ,,  &a  x      ..    .    f  if  . 

i  .ds  —  dh  —  —a  ---  -  .  dh  +  -  ---  ds. 
g  A  m  2g 

Therefore 

i-L±     i-t*-. 

dh  m  2g  m  2gi 

~3s=          ~ur^=l~         u'x  .....     (^ 
I  —  a-  I  —  a— 

gA  gA 

Let  the  position  of  any  point  a  in  the  surface  be  defined  by 
its  perpendicular  distance  h  from  the  bed  and  by  the  distance  s 
of  the  transverse  section  at  a  from  an  origin  in  the  bed.  Then 

r  is  the  tangent  of  the  angle  which  the  tangent  to  the  surface 
ds 


FLOW  OF   WATER  IN  OPEN  CHANNELS.  l6l 

at  a  makes  with  the  bed.  It  is  positive  or  negative  according 
as  the  depth  increases  or  diminishes  in  the  direction  of  flow, 
thus  defining  two  states  of  steady  varied  motion. 

Between  these  there  is  an  intermediate  state  defined  by 

dh  f  u* 

_-  =  o  =  * «-  ^-  — , 
as  m2g 

f  u* 

and  i  =  —  —    is  the  equation  for  steady  flow  with  uniform 
m2g 

motion. 

Let  £/",  M,  H  be  the  corresponding  values  of  #,  m,  h  in  the 
case  of  uniform  motion.     Then 


and  equation  5  becomes 


__ 
dh  m  U* 


I  —  a 

EXAMPLE. — Consider  a  stream  of  rectangular  section  and 
of  a  width  x  which  is  very  great  as  compared  with  the  depth. 
Then 

A  =  xh  ;  P  =  x  very  nearly ;    m  =  --  =-  h ;     M  =  -  -  =  H. 
Hence 

*-TW        '-(T)' 

/*   £7*  \  h  I 


dh 


I  —  a—f  I  —  a-- 

gh  gh 


since  xhu  =  xHU  and  therefore  —  -.  =  -r-. 

£/        h 

Note.  —  In  each  of  the  following  cases  the  line  PQ  drawn 


1 62  H  YD RA  ULICS. 

parallel  to  the  bed,  represents  the  surface  of  uniform  motion, 
H  being  the  distance  between  PQ  and  the  bed. 
CASE  I.     au*  <  gh     and     H  <  h. 

—  is  positive,  and  therefore  h  increases  in  the  direction  of 
as 

flow.     Thus  the  actual  surface  MN  of  the  stream  is  wholly 
above  the  line  PQ. 


FIG.  94. 

Proceeding  up  stream,  h  becomes  more  and  more  nearly 
equal  to  H,  so  that  the  numerator  of  equation  8,  and  therefore 

also  — -,  approximates  more  and  more  closely  to  zero, 
as 

Again,  proceeding  down-stream,  h  increases  and  u  dimin- 
ishes, so  that  the  numerator  and  denominator  in  equation  8 
approximate  each  more  and  more  closely  to  the  value  unity, 

and  therefore  —  becomes  more  and  more  nearly  equal  to  i, 
as 

the  slope  corresponding  to  uniform  motion. 

Hence  up-stream,  MN  is  asymptotic  to  PQ,  and  down- 
stream MN  is  asymptotic  to  a  horizontal  line.  This  form  of 
water-surface  is  produced  when  a  weir  is  built  across  a  channel 
in  which  the  water  had  previously  flowed  with  a  uniform 
motion. 

CASE  II.     au*  <  gh     and     H  >  h. 

—  is  now  negative,  and  the  depth  diminishes  in  the  direc- 
ds 

tion  of  flow. 

Up-stream,  h  increases  and  approaches  H  in  value,  so  that 
MN  is  asymptotic  to  PQ. 


FLOW  OF   WATER  IN  OPEN  CHANNELS. 


163 


Down-stream,  h  diminishes,  u  increases,  and  therefore  the 

value  of  —  is  more  and  more  nearly  equal  to  unity, 
gh 

Thus,  in  the  limit,  the  denominator  in  equation  8  becomes 

zero,  and  therefore  —  =  00.     Hence  theory  indicates  that  at  a 
as 

certain  point  down-stream  the  surface  line  MN  takes  a  direc- 
tion which  is  at  right  angles  to  the  general  direction  of  flow. 
This  is  contrary  to  the  fundamental  hypothesis  that  the  fluid 
filaments  flow  in  sensibly  parallel  lines.  In  fact,  before  the 


FIG.  95. 

limit  could  be  reached  this  hypothesis  would  cease  to  be  even 
approximately  true,  and  the  general  equation  would  cease  to 
be  applicable.  This  form  of  water-surface  is  produced  when 
there  is  an  abrupt  depression  in  the  bed  of  the  stream. 

Fig.  96  shows  one  of  the  abrupt  falls  in  the  Ganges  canal 
as  at  first  constructed.     The  surface  of  the  water  flowing  freely 


FIG.  96. 

over  the  crest  of  the  fall  took  a  form  similar  to  MN  below  the 
line  PQ.oi  uniform  motion.  The  diminution  of  depth  in  the 
approach  to  the  fall  caused  an  increase  in  the  velocity  of  flow, 
with  the  result  that  for  several  miles  above  the  fall  a  serious 


164 


HYDRA  ULICS. 


erosion  of  the  bed  and  sides  took  place.  In  order  to  remedy 
this,  temporary  weirs  were  constructed  so  as  to  raise  the  level 
of  the  water  until  the  surface-line  assumed  a  form  MN'  cor- 
responding approximately  to  PQ.  In  some  cases  the  water 
was  raised  above  its  normal  height  and  a  backwater  produced* 
CASE  III.  au*  >  gh  and  H  <  h. 

—-  is  negative  and  the  surface-line  of  the  stream  is  wholly 
above  PQ. 


FIG.  97. 


dk 


If  h  gradually  increases,  u  diminishes  and  —j-  approximates 

to  —  i  in  value. 

If  h  gradually  diminishes  it  approximates  to  H  in  value, 

dk 

and  in  the  limit  -T~=  o. 
ds 

Between  these  two  extremes  there  is  a  value  of  h  for  which 
the  denominator  of  equation  8  becomes  nil,  viz., 


and  the  corresponding  value  of  -y-  is  infinity. 

Thus  one  part  of  the  surface  line  is  asymptotic  to  PQ,  the 
line  of  uniform  motion,  another  part  is  asymptotic  to  a  hori- 
zontal line,  while  at  a  certain  point  at  which  the  depth  is 


the  surface  of  the  stream  is  normal  to  the  bed. 


FLOW  OF   WATER  IN  OPEN  CHANNELS. 


I6S 


This  is  contrary  to  the  fundamental  hypothesis  that  the 
fluid  filaments  flow  in  sensibly  parallel  lines,  and  the  general 
equation  no  longer  represents  the  true  condition  of  flow. 

In  cases  such  as  this,  there  has  been  an  abrupt  rise  of  the 
surface  of  the  stream,  and  what  is  called  a  "  standing  wave  " 
has  been  produced. 

In  a  stream  of  depth  H  flowing  with  a   uniform  velocity 

tgr 

•depth  to  h^  which  is  > 


U  which  is  >  \  /  — — ,  construct  a  weir  so  as  to  increase  the 

all* 


Then  in  one  portion  of  the  stream  near  the  weir  the  depth 

aU*  aU* 

is  >  ,  while  further  up  the  stream  the  depth  is  <  — — . 

o  o 

U* 
Thus  at  some  intermediate  point  the  depth  =  a ,  the  cor- 

o 


dh 


responding  value  of  -r-  being  oo ,  so  that  at  this  point  a  stand- 


ds 


ing  wave  is  produced. 
Now 


flT 


=  Mi=-Hi\ 


and  since 


1  66 


HYDRAULICS. 


and  therefore 


which  condition  must  be  fulfilled  for  a  standing  wave. 
Bazin  gives  the  following  table  of  values  of/: 


Nature  of  Bed. 

Slope  (A  =  /) 

below  which  stand- 
ing wave  is  im- 
possible.    In 
Metres  per  Metre. 

Standing  Wave  Produced. 

Slope  in  Metres 
per  Metre  (or 
Feet  per  Foot). 

Least  Depth 
in  Metres. 

Very  smooth  cemented  surface  

.00147 
.00186 
.00235 

•00275 

{.002 
.003 
.004 
(  -003 
4  .004 
(  .006 
.004 
•    .006 

.010 

.006 
•    .010 
.015 

.08 
•03 
.02 
.12 
.06 
•03 
.36' 
.16 
.08 
I.  O6 
•  47 
.28 

Earth                             

A  standing  wave  rarely  occurs  in  channels  with  earthen 
beds,  as  their  slope  is  almost  always  less  than  the  limit,  .00275. 

The  formation  of  a  standing  wave  was  first  observed  by 
Bidone  in  a  small  masonry  canal  of  rectangular  section. 

The  width  of  the  canal  =  0^.325  =  x  • 

"    slope  f=  -j)  of  the  canal  —       -023  » 

"    uniform  velocity  of  flow     =  1^.69    =  U\ 
"    depth  corresponding  to  U  =  0^.064  =  H. 
A  weir  built  across  the  canal  increased  the  depth  of  the 
water  near  the  weir  to  ow.287  =  h^ 

It  was  found  that  the  "  uniform  regime  "  was  maintained 
up  to  a  point  within  4^.5  of  the  weir.  At  this  point  the 
depth  suddenly  increased  from  0^.064  to  about  ow.i7O,  and 
between  the  point  and  the  weir  the  surface  of  the  stream  was 
slightly  convex  in  form  (Fig.  98). 


FLOW  OF   WATER   IN   OPEN   CHANNELS. 


i67 


With  the  preceding  data  and  taking  a  =  i.i, 

is  therefore  >  I  at  a  section  ab,  Fig.  99. 
At  the  section  cd, 


=q 


H_ 

h 


.064 
^87 


X  1.69  =  0^.377, 


and 


=  .055  and  is  therefore  <  i. 


au 


FIG.  99* 


Thus  the  expression  I  --  —is  negative  for  a  section  ao 

and  positive  for  a  section  cdt  so 
that  z  must  change  sign  between 

these  sections,  and  —  will  then 
as 

become  infinite. 

Consider  a  portion  of  a 
stream  bounded  by  two-  trans- 
verse sections  ab,  cd,  in  which  a  standing  wave  occurs,  Fig.  99. 

Assume  that  the  fluid  filaments  flow  across  the  sections  in 
sensibly  parallel  lines. 

Let  the  velocities  and  area  at  section  ab  be  distinguished 
by  the  suffix  i,  and  those  at  cd\sy  the  suffix  2.     Then 
Change  of  momentum    in  di-  ) 

rection  of  flow  [  ==  imPulse  in  same  direction. 

Hence 


w 
— 


and  therefore 


=A1yl  -  A,vv      ...     (9) 


the  depths  below  the  surface  of  the  centres  of 
gravity  of  the  sections  ab,  cd,  respectively. 


1  68  H  YDRA  ULICS. 

Now,  vl  =  ul  +  Vr     Therefore 


Also,  as  already  shown, 

a,A,U:  =  2av?  =  AM' 
and,  neglecting  F,  as  compared  with  3«,  , 

**#•  =-Arf  + 

Thus 


and  hence 

ufA-i, 

=  ~^-L(a  +  2)  =  aA*u» 

0 

a  4-  2 
where  a'  =  — ! — ,  and  is   1.033  «  *  —  !•!• 

Similarly  it  may  be  shown  that 


Thus  equation  9  becomes 


~(A^  -  Ap?)  =  ^^  -  Aj,.     .     .     .    (10) 

Let  the  section  of  the  canal  be  a  rectangle  of  depth  Hl  at 
ab  and  Ht  at  ^.     Then 

ff  H 

ufr  =  u,H,  ;     -±  =  >,  ;     -y-=  ^. 


FLOW  OF   WATER   IN   OPEN   CHANNELS.  169 

Therefore,  by  equation  10, 


which  reduces  to 


//,  =  H^  satisfies  the  equation  and  corresponds  to  a  condition 
of  uniform  motion. 
Also 

a'u?  ^ff.ff.  +  ff, 

g         Hl        2 

In  Bidone's  canal,  u1  =  1^.69,  Hl  =  0^.064.  Substituting 
these  values  in  equation  II,  the  value  of  H^  is  found  to  be 
ow.  16,  which  agrees  somewhat  closely  with  the  actual  measure- 
ments. 

N.B. — The  coefficients  a  and  a'  have  not  been  very  accu- 
rately determined,  but  their  exact  values  are  not  of  great 
importance.  They  are  often  taken  equal  to  unity. 


H YD RA  ULICS. 


EXAMPLES. 

1.  What  fall  must  be  given  to  a  canal  2600  ft.  long,  7  ft.  wide  at  the 
top,  3  ft.  wide  at  the  bottom,  \\  ft.  deep,  and  conveying  40  cubic  ft.  of 
water  per  second  ?    /=^¥.  Ans.   i  in  135. 

2.  Determine  the  fall  of  a  canal  1500  ft.  long,  of  2  ft.   lower,  8  ft. 
upper  breadth,  and  4  ft.  deep,  which  is  to  convey  70  cubic  feet  of  water 
per  second.  Ans.   i  in  1365.4. 

3.  For  a  distance  of  300  ft.  a  brook  with  a  mean  water  perimeter  of 
40  ft.  has  a  fall  of  9.6  in.;  the  area  of  the  upper  transverse  profile  is  70 
sq.  ft.,  that  of  the  lower  60  sq.  ft.     Find  the  discharge. 

Ans.  662.87  cub.  ft.  per  sec. 

4.  In  a  horizontal  trench  5  ft.  broad  and  800  ft.  long  it  is  desired  to 
carry  off  20  cub.  ft.  discharge  and  to  let  it  flow  in  at  a  depth  of  2  ft. ; 
what  must  be  the  depth  at  the  end  of  the  canal  ?     (/  =  .008.) 

Ans.  1.64  ft. 

5.  Water  flows  along  an  open  channel  12  ft.  wide  and  4  ft.  deep,  at 
the  rate  of  2  ft.  per  second.     What  is  the  fall?     A  dam  12  ft.  by  3  ft. 
high  is  formed  across  the  channel;  how  high  will  the  water  rise  over  the 
crest  of  the  dam  ?  Ans.   i  in  48o,/  being  .08  ;  .899  ft. 

6.  A  stream  is  rectangular  in  section,  12  ft.  wide,  4  ft.  deep,  and  falls 
i  in  100.     Determine  the  discharge  (i)  with  an  air-perimeter;  (2)  without 
air-perimeter.  Ans.  (i)  645.398  cub.  ft.  per  sec. 

(2)  665.088  cub.  ft.  per  sec. 

7.  A  canal  20  ft.  wide  at  the  bottom  and  having  side  slopes  of  i£  to 
i  has  8  ft.  of  water  in  it;  find  the  hydraulic  mean  depth.  Ans.  5.24  ft. 

8.  The  water  in  a  semicircular  channel  of  10  ft.  'radius,  when  full 
flows  with  a  velocity  of  2  ft.  per  second  ;  the  fall  is  i  in  400.    Find  the  co- 
efficient of  friction.  Ans.  .2. 

9.  Calculate  the  flow  per  minute  across  a  given  section  of  a  rectarw- 
gular  canal  20  ft.  deep,  45  ft.  wide,  the  slope  of  the  bed  being  22  in.  per 
mile  and  the  coefficient  of  friction  per  square  foot  =  .008. 

Ans.  279,229  cub.  ft. 

10.  Why  does  the  water  of  the  St.  Lawrence  rise  on  the  formation 
of  the  ice  ? 

11.  Find  the  depth  and  width  of  a  rectangular  stream  of  900  sq.  ft. 
sectional  area,  so  that  the  flow  might  be  a  maximum  ;  also  find  the  flow, 

f  being  .008  and  the  slope  22  in.  per  mile. 

Ans.  21.21  ft.;  42.42  ft.;  4885  cub.  ft.  per  second. 


FLOW  OF   WATER  IN  OPEN  CHANNELS.  \7\ 

12.  Water  flows  along  a  symmetrical  channel,  20  ft.  wide  at  top  and 
8  ft.  wide  at  bottom  ;  the  friction  at  the  sides  varies  as  the  square  of  the 
velocity,  and  is  i  Ib.  per  square  foot  for  a  velocity  of  16  ft.  per  second. 
Find  the  proper  slope,  so  that  the  water  may  flow  at  the  rate  of  2  ft.  per 
second  when  its  depth  is  6  ft.  Arts,   i  in  3445. 

13.  Calculate  the  flow  across  the  vertical  section  of  a  stream  4  ft. 
deep,  1 8  ft.  wide  at  top,  6  ft.  wide  at  bottom,  the  slope  of  the  surface 
being  18  in.  per  mile.     (/=  .008.)       Ans.  110.9376  cub.  ft.  per  second. 

14.  The  sewers  in  Vancouver  are  square  in  section  and  are  laid  with 
one  diagonal   vertical.     To  what  height  should  the  water  rise  so  that 
(a)  the  velocity  of  flow  may  be  a  maximum  ;   (b)  the  discharge  may  be  a 
maximum  ?     (A  side  of  the  square  =  12  in.)' 

Ans.  (a)  .292  ft.  above  horizontal  diameter. 
(b)  .5797  ft.    " 

15.  The  sides  of  an  open  channel  of  given  inclination  slope  at  45* 
and  the  bottom  width  is  20  ft.    Find  the  depth  of  water  which  will  make 
the  velocity  of  flow  across  a  vertical  section  a  maximum. 

Ans.  6.73  ft. 

17.  The  banks  of  a  channel  slope  at  45° ;  the  flow  across  a  transverse 
section  is  to  be  at  the  rate  of  100  cubic  feet  at  a  maximum  velocity  of  5 
ft.  per  second.     Determine  the  dimensions  of  the  transverse  profile. 

Ans.   11.05  ft.  wide  at  bottom  ;  2.28  ft.  deep. 

1 8.  What  dimensions  must  be  given  to  the  transverse  profile  of  a 
canal  whose  banks  slope  at  40°,  and  which  has  to  conduct  away  75  cubic 
feet  with  a  mean  velocity  of  3  ft.  per  second  ? 

Ans.  Depth  =  3.6  ft. ;  width  at  bottom  =  2.62  ft. 

19.  The  section   of  a  canal   is  a  regular  trapezoid ;  its  slope  is  i  in 
500 ;  its  width  at  the  bottom  is  8  ft.;  the  sides  are  inclined  at  30°  to  the 
vertical.     On   one  occasion  when  the  water  was  4  ft.  deep  a  wind  was 
blowing  up  the  canal,  causing  an  air-resistance  for  each  unit  of  free  sur- 
face equal  to  one  fifth  of  that  for  like  units  at  the  bottom  and  sides, 
where  the  coefficient  of  friction  may  be  taken  to  be  .08. 

Determine  the  discharge.     How  will  the  discharge  be  affected  when 
the  canal  is  frozen  over?  Ans.  75.34  cub.  ft.  per  sec. 

20.  The  section  of  a  channel  is  a  rhombus  with  diagonal  vertical. 
How  high  must  the  water  rise  in  the  channel  (a)  to  give  a  maximum  of 
flow,  and  (b)  to  give  a  maximum  discharge? 

Ans.  If  D  is  the  length  of  the  horizontal  diameter,  and  if  & 
is  the  inclination  of  a  side  to  the  vertical,  the  water 
must  rise  above  the  horizontal  diameter  to  the  height 
Z)cot0  x  .207  in  (a)  and  to  the  height  Z>cotfl  x  .4099 
in  (b). 

21.  In  the  transverse  section  ABCD  of  an  open  channel  with  a  verti- 
cal slope  of  i  in  300,  the  bottom  width  is  20  ft.,  the  angle  ABC  —  90* 


1 72  H  YDRA  ULICS. 

and  the  angle  BCD  =  45°.  Find  the  height  to  which  the  water  will 
rise  so  that  the  velocity  of  flow  may  be  a  maximum  ;  also  find  the  dis- 
charge across  the  section,/  being  .008. 

Ans.   11.715  ft.;  1584  cub.  ft.  per  second. 

22.  A  canal  is  20  ft.  wide  at  the  bottom,  its  side  slopes  are  i|  to  i,  its 
longitudinal  slope  is  i  in  360;  calculate  H.M.D.  and  the  flow  per  minute 
across  any  given  vertical  section  when  there  is  a  depth  of  8  ft.  of  water 
in  the  canal.     (Coeff.  of  friction  =  .008.) 

Ans.  5.24  ft.;  2762.7776  cub.  ft.  per  second. 

23.  If  a  weir  2  ft.  high  were  built  across  the  canal  in  the  preceding 
question,  what  would  be  the  increase  in  the  depth  of  the  water? 

Ans.  2.79  ft. 

24.  For  a  small  tachometer  the  velocities  are  .163,  .205,  .298,  .366, 
,61    metre;  the   number  of  revolutions  per  second  are  .6,  .835,  1.467, 
1.805,  3.142.     Find  the  constants  corresponding  to  the  wheel. 

Ans.  ,162;  .202;  .309;  .367;  .595. 

25.  If  the  head  of  water  in  a  channel  increase  by  one  tenth,  show 
that  the  velocity  and  discharge,  respectively,  increase  by  -£$  and   ^. 
approximately. 

If  the  depth  diminish  by  8$,  show  that  the  velocity  and  discharge, 
respectively,  diminish  by  4%  and  12%,  approximately. 

26.  Assuming  (i)  that  a  river  flows  over  a  bed  of  uniform  resistance 
to  source ;  (2)  that  to  maintain  stability  the  velocity  is  constant  from 
source  to  mouth ;   (3)  that  the  river  sections  at  all  points  are  similar  ; 
(4)  that  the  discharge  increases  uniformly  in  consequence  of  the  supply 
from  affluents — determine  the  longitudinal  section  of  such  a  river. 

Ans.  A  parabola. 


CHAPTER  V. 


METHODS  OF  GAUGING. 

I.  Gauging    of    Streams    and    Watercourses.  —  The 

amount  of  flow  Q  in  cubic  feet  per  second  across  a  transverse 
section  of  A  sq.  ft.  in  area  is  given  by  the  expression 


Q  -  Au, 

u  being  the  mean  velocity  of  flow  in  the  section  in 
feet  per  second.  Various  methods  are  employed  for 
the  determination  of  u. 

METHOD  I.  The  most  convenient  method  for 
gauging  small  streams,  canals,  etc.,  is  by  means  of 
a  temporarily  constructed  weir,  which  usually  takes 
the  form  of  a  rectangular  notch.  The  greatest 
care  should  be  exercised  to  ensure  that  the  crest 
of  the  weir  is  truly  level  and  properly  formed  and 
that  the  sides  are  truly  vertical.  The  difference  of 
level  between  the  crest  of  the  weir  and  the  surface 
of  the  water  at  a  point  where  it  has  not  begun  to 
slope  down  towards  the  weir  is  best  es- 
timated by  means  of  Boyden's  hook  gauge, 
Fig.  100. 

This  gauge  consists  of  a  carefully  grad- 
uated rod,  or  of  a  rod  with  a  scale  attached, 
having  at  the  lower  end  a  hook  with  a  thin 
flat  body  and  a  fine  point.  The  rod  slides 
in  vertical  supports,  and  a  slow  vertical 
movement  is  given  by  means  of  a  screw  of 
fine  pitch.  In  an  experiment,  the  hook 


FIG.  TOO. 


point  is  set  truly  level  with  the  crest  of  the  weir,  and  a  read- 
ing is  taken.     The  gauge  is  then  moved  away  from  the  weir, 


HYDRA  ULICS. 


about  2  to  4  ft.  for  small  weirs  and  about  6  to  8  ft.  for  large 
weirs.  The  hook  is  then  slowly  raised,  until  a  capillary  eleva- 
tion of  the  surface  is  produced  over  the  point.  The  hook  is 
now  lowered  until  this  elevation  is  barely  perceptible,  and  a 
second  reading  is  taken.  The  difference  between  the  two 
readings  is  the  difference  of  level  required. 

In  ordinary  light,  differences  of  level  as  small  as  the  one- 
thousandth  of  a  foot,  can  be  easily  detected  by  the  hook 
gauge,  while  with  a  favourable  light  it  is  said  that  an  experi- 
enced observer  can  detect  a  difference  of  two  ten-thousandths 
of  a  foot. 

METHOD  II.  A  portion  of  the  stream  which  is  tolerably 
straight  and  of  approximately  uniform  section  is  defined  by 
two  transverse  lines  O^B,  OfD,  at  any  distance  5  ft.  apart. 


FIG.  101. 

The  base-line  O,O^  is  parallel  to  the  thread  EF  of  the 
stream,  and  observers  with  chronometers  and  theodolites  (or 
sextants)  are  stationed  at  (9, ,  <92.  The  time  T  and  path  EF 
taken  by  a  float  between  AB  and  CD  can  now  be  determined. 
At  the  moment  the  float  leaves  A B  the  observer  at  Ol  signals 
the  observer  at  (92,  who  measures  the  angle  O^O^E,  and  each 
marks  the  time.  On  reaching  CD  the  observer  at  O.t  signals 
the  observer  at  Ol ,  who  measures  the  angle  O^Of,  and  each 
again  marks  the  time. 

Experience  alone  can  guide  the  observer  in  fixing  the  dis- 


METHODS   OF  GAUGING. 


175 


tance  5  between  the  points  of  observation.  It  should  be 
remembered  that  although  the  errors  of  time  observations  are 
diminished  by  increasing  S,  the  errors  due  to  a  deviation  from 
lines  parallel  to  the  thread  of  the  stream  are  increased. 

A  number  of  floats  may  be  sent  along  the  same  path,  and 

their  velocities  UsJ  are  often  found  to  vary  as  much  as  20  per 

cent  and  even  more. 

Having  thus  found  the  velocities  along  any  required  num- 
ber of  paths  in  the  width  of  the  stream,  the  mean  velocity  for 
the  whole  width  can  be  at  once  determined. 

Surface-floats  are  small  pieces  of  wood,  cork,  or  balls  of 
wax,  hollow  metal  and  wood,  colored  so  as  to  be  clearly  seen, 
and  ballasted  so  as  to  float  nearly  flush  with  the  water-surface 
and  to  be  little  affected  by  the  wind. 

Subsurface-floats. — A  subsurface-float  consists  of  a  heavy 
float  with  a  comparatively  large  intercepting  area,  maintained 
at  any  required  depth  by  means  of  a  very  fine  and  nearly 
vertical  cord  attached  to  a  suitable  surface-float  of  minimum 
immersion  and  resistance.  Fig.  102  shows  such  a  combina- 
tion, the  lower  float  consisting  of  two  pieces  of  galvanized  iron 
soldered  together  at  right  angles,  the  upper  float  being  merely 
a  wooden  ball. 


FIG.  102. 


FIG.  103. 


Another  combination  of  a  hollow  metal  ball  with  a  piece 
of  cork  is  shown  by  Fig.  103. 

The  motion  of  the  combination  is  sensibly  the  same  as  that 


HYDRA  ULICS. 


of  the  submerged  float,  and  gives  the  velocity  at  the  depth  to 
which  the  heavy  float  is  submerged. 

Twin-floats. — Two  equal  and  similar  floats  (Fig.  104),  one 
denser  and  the  other  less  dense  than  water, 
1  are  connected  by  a  fine  cord.  The  velocity 
(vt)  of  the  combination  is  approximately  the 
mean  of  the  surface-velocity  (vs)  and  of  the 
velocity  (v^)  at  the  depth  to  which  the  heavier 
float  is  submerged.  Thus 


FIG.  104.        and  therefore 

d>  ~~"~   ^    t  */s  9 

so  that  vd  can  be  determined  as  soon  as  the  value  of  vt  has 
been  observed  and  the  value  of  vs  found  by  surface-floats. 

Velocity-rod.  —  This  is  a  hollow  cylindrical  rod  of  ad- 
justable leiigth  and  ballasted  so  as  to  float  nearly  vertical.  It 
sinks  almost  to  the  bed  of  the  stream, 
and  its  velocity  (vm)  is  approximately  the 
mean  velocity  for  the  whole  depth. 

Francis  gives  the  following  empirical 
formula  connecting  the  mean  velocity 
(vm)  with  the  observed  velocity  (vr)  of 
the  rod : 

...*/£), 


=zv(i.oi2 


d  being  depth  of  stream,  and  d'  the  depth  FlG-  I05- 

of  water  below  bottom  of  rod  ;  but  d'  should  not  exceed  about 

one  fourth  of  d. 

METHOD  III.  Pitot  Tube  and  Darcy  Gauge.— A  Pitot 
tube  (Figs.  106  to  108)  in  its  simplest  form  is  a  glass  tube  with 
a  right-angled  bend.  When  the  tube  is  plunged  vertically  into 
the  stream  to  any  required  depth  z  below  the  free  surface,  with 
its  mouth  pointing  up-stream  and  normal  to  the  direction  of 


METHODS   OF  GAUGING. 


177 


flow,  the  water  rises  in  the  tube  to  a  height  h  above  the  out- 
side surface,  and  the  weight  of  the  column   of  water  z  -f-  h 


FIG.  i 06. 


FIG.  107. 


FIG.  108. 


high,  is  balanced  by  the  impact  of  the  stream  on  the  mouth. 
Hence,  (Chap.  VI.), 


wA(z  -f-  k)  =  wAz  -f-  kwA  —  , 


and  therefore 


A  being  the  sectional  area  of  the  tube,  u  the  velocity  of  flow 
at  the  given  depth,  and  k  a  coefficient  to  be  determined  by 
experiment. 

A  mean  value  of  k  is  1.19.  With  a  funnel-mouth  or  a  bell- 
mouth,  Pitot  found  k  to  be  1.5.  This  form  of  mouth,  however, 
interferes  with  the  stream-lines,  and  the  velocity  in  front  of 
the  mouth  is  probably  a  little  different  from  that  in  the  unob- 
structed stream. 

The  advantages  of  tubes  of  small  section  are  that  the  dis- 
turbance of  the  stream-lines  is  diminished  and  the  oscillations 
of  the  column  of  water  are  checked.  Darcy  found  by  careful 
measurement  that  the  difference  of  level  between  the  surfaces 
of  the  water-column  in  a  tube  of  small  section  placed  as  in 
Fig.  106,  and  of  the  water-column  placed  as  in  Fig.  107  with 


HYDRA  ULICS. 


FIG.  109. 


its  mouth  parallel  to  the 
direction  of  flow,  is  almost 

exactly  equal  to  — -. 

When  the  tube  is  placed 
as  in  Fig.  108  with  its 
mouth  pointing  down- 
stream and  normal  to  the 
direction  of  flow,  the  level 
of  the  surface  of  the  water 
in  the  tube  is  at  a  depth  ti 
below  the  outside  surface, 
and 


where  kf  is  a  coefficient  to 
be  determined  by  experi- 
ment and  a  little  less  than 
unity. 

In  this  case  the  tube 
again  obstructs  the  stream- 
lines. Pitot's  tube  does 
not  give  measurable  indi- 
cations of  very  low  veloc- 
ities. A  serious  objection 
to  the  simple  Pitot  tube  is 
the  difficulty  of  obtaining 
accurate  readings  near  the 
surface  of  the  stream.  This 
objection  is  removed  in 
the  case  of  Darcy's  gauge, 
shown  in  the  accompany- 
ing sketch,  Fig.  109. 

A  and  B  are  the  water- 
inlets;  C  and  D  are  two 
double  tubes  ;  E  is  a  brass 


METHODS   OF  GAUGING. 

tube  containing  two  glass  pipes  which  communicate  at  the 
bottom  with  the  water-inlets  and  at  the  top  with  each  other, 
and  with  a  pump  F  by  which  the  air  can  be  drawn  out  of 
the  glass,  pipes  thus  allowing  the  water  to  rise  in  them  to  any 
convenient  height. 

Thus  Darcy's  gauge  really  consists  of  two  Pitot  tubes  con- 
nected by  a  bent  tube  at  the  top  and  having  their  mouths  at 
right  angles  or  pointing  in  opposite  directions.  If  h  is  the 
difference  of  level  between  the  water-surfaces  in  the  tubes 
when  the  mouths  are  at  right  angles,  then 


and   Darcy's   experiments  showed   that   k  does   not  sensibly 
differ  from  unity. 

When  the  mouths  point  in  opposite  directions,  let  h^  h^  be 
the  differences  of  level  between  the  stream-surface  and  the 
surfaces  of  the  water  in  the  tube  pointing  up-stream  and  the 
tube  pointing  downstream,  respectively.  Then 

u* 

**  =  k{2g'> 
U* 

and  therefore 

u* 

h  j.  h  -  (k ,  +  k  )— 

*>2g 


where  k  =  kv  -\-  k^ 

k  having  been  determined  experimentally  once  for  all,  the 
difference  of  level  (=  h^  -\-  h^)  between  the  columns  for  any 
given  case  can  be  measured  on  the  gauge  and  then  u  can  be 
at  once  found. 


1 80  H  YDRA  ULICS. 

A  cock  may  be  inserted  in  the  bend  connecting  the  two 
tubes,  and  through  this  cock  air  may  be  exhausted  and  a 
partial  vacuum  created  in  the  upper  portion  of  the  gauge. 
The  water-columns  will  thus  rise  to  higher  levels,  but  the  dif- 
ference between  them  will  remain  constant.  Thus  the  surface 
of  the  column  in  the  down-stream  tube  may  be  brought  above 
the  level  of  the  outside  surface,  and  the  reading  is  then  easily 
made. 

Sometimes  the  gauge  is  furnished  with  cocks  at  the  lower 
parts  of  the  tubes,  and  if  these  cocks  are  closed  when  the 
measurement  is  to  be  made,  the  gauge  may  be  removed  from 
the  stream  for  the  readings  to  be  taken. 

METHOD  IV.  Current-meters. — The  velocity  of  flow  in 
large  streams  and  rivers  is  most  conveniently  and  most  ac- 
curately ascertained  by  means  of  the  current-meter.  The 
earliest  form  of  meter,  the  Woltmann  mill,  is  merely  a  water- 
mill  with  flat  vanes,  similar  in  theory  and  action  to  the  .wind- 
mill. When  the  Woltmann  is  plunged  into  a  current,  a  counter 
registers  the  number  of  revolutions  made  in  a  given  interval 
of  time,  and  the  corresponding  velocity  can  then  be  deter- 
mined. This  form  of  meter  has  gone  out  of  use  and  has  been 
replaced  by  a  variety  of  meters  of  greater  accuracy,  of  finer 
construction,  and  much  better  suited  to  the  work.  In  its  sim- 
plest form  the  present  meter  consists  of  a  screw-propeller 
wheel  (Fig.  1 10),  or  a  wheel  with  three  or  more  vanes  mounted 
on  a  spindle  and  connected  by  a  screw-gearing  with  a  counter 
which  registers  the  number  of  revolutions.  The  meter  is  put 
'  in  or  out  of  gear  by  means  of  a  string  or  wire.  When  a  cur- 
rent velocity  at  any  given  point  is  to  be  found,  the  reading  of 
the  counter  is  noted,  the  meter  is  sunk  to  the  required  position, 
and  is  then  set  and  kept  in  gear  for  any  specified  interval  of 
time.  At  the  end  of  the  interval  the  meter  is  put  out  of  gear 
and  is  raised  to  the  surface  when  the  reading  of  the  counter  is 
again  noted.  The  difference  between  the  readings  gives  the 
number  of  revolutions  made  during  the  interval,  and  the  veloc- 
ity is  given  by  an  empirical  formula  connecting  the  velocity 
and  the  number  of  revolutions  in  a  unit  of  time. 


METHODS   OF  GAUGING. 


The  vane  Fis  introduced  to  compel  the  meter  to  take  its 
proper  direction. 

In  order  to  prevent  the  mechanism  of  the  meter  from  being 


FIG.  1 10. 


FIG.  in. 


injuriously  affected  by  floating  particles  of  detritus,  Revy  en- 
closed vthe  counter  in  a  brass  box,  Fig.  ill,  with  a  glass  face, 


FIG.  112. 


FIG.  ri3. 

and  filled  the  box  with  pure  water  so  as  to  ensure  a  constant 
coefficient  of  friction  for  the  parts  which  rub  against  each 
other.  In  the  best  meters,  however,  the  record  of  the  number 


1  82  HYDRAULICS. 

of  revolutions  is  kept  by  means  of  an  electric  circuit,  Fig.  112, 
which  is  made  and  broken  once,  or  more  frequently,  each 
revolution,  and  which  actuates  the  recording  apparatus.  The 
time  at  which  an  experiment  begins  and  ends  is  noted,  and  the 
revolutions  made  in  the  interval  are  read  on  the  counter,  which 
may  be  kept  in  a  boat  or  on  the  shore,  as  the  circumstances  of 
the  case  may  require.  The  meter  is  usually  attached  to  a  suit- 
ably graduated  pole,  so  that  the  depth  of  the  meter  below  the 
water-surface  can  be  directly  read.  The  mean  velocity  for  the 
whole  depth  at  any  point  of  a  stream  may  be  found  by  moving 
the  meter  vertically  down  and  then  up,  at  a  uniform  rate. 
The  mean  of  the  readings  at  the  two  surface  positions  and  at 
the  bottom  position  will  be  the  number  of  revolutions  corre- 
sponding to  the  mean  velocity  required.  The  mean  velocity 
for  the  whole  cross-section  may  also  be  determined  by  moving 
the  meter  uniformly  over  all  parts  of  the  section. 

Before  the  meter  can  be  used  it  must  be  rated.  This  is 
done  by  driving  the  meter  at  different  uniform  speeds  through 
still  water.  Experiment  shows  that  the  velocity  v  and  the 
number  of  revolutions  n  are  approximately  connected  by  the 
formula 

v  =  an  +  b, 

where  a  and  b  are  coefficients  to  be  determined  by  the  method 
of  least  squares  or  otherwise. 
Exner  gives  the  formula 


VQ  being  the  velocity  at  which  the  meter  just  ceases  to  re- 
volve. 

OTHER  METHODS.  —  Many  other  pieces  of  apparatus  for 
the  measurement  of  current  velocities  have  been  designed. 

PerrodiTs  hydrodynamometer,  for  example,  gives  the  ve- 
locity directly  in  terms  of  the  angle  through  which  a  vertical 
torsion-rod  is  twisted,  and  in  this  respect  is  superior  to  the 
current-meter. 


METHODS   OF  GAUGING. 


183 


FIG.  114. 


The  hydrometric  pendulum  (Fig.  114),  again,  connects  the 
velocity  with  the  angular  devia- 
tion from  the  vertical  of  a  heavy 
ball  suspended  by  a  string  in  the 
current. 

Hydrometric  and  torsion  bal- 
ances have  also  been  devised, 
but  they  must  be  regarded 
rather  as  curiosities  than  as 
being  of  any  real  practical  use. 

2.  Gauging  of  Pipe  Flow. — A  variety  of  meters  have 
been  designed  to  register  the  quantity  of  water  delivered  by  a 
pipe.  The  principal  requisites  of  such  a  meter  are : 

1.  That  it   should   register  with  accuracy  the  quantity  of 
water  delivered  under  different  pressures. 

2.  That  it   should    not  appreciably  diminish  the   effective 
pressure  of  the  water. 

3.  That    it    should    be    compact    and    adaptable   to    every 
situation. 

4.  That  it  should  be  simple  and  durable. 

The  Venturi  Meter  (Fig.  115)  is  so  called  from  Venturi,  who 
first  pointed  out  the  relation  between  the  pressures  and  veloci- 
ties of  flow  in  converging  and  diverging  tubes. 


FIG.  115. 

As  shown  by  the  longitudinal  section,  Fig.  116,  this  meter 
consists  of  two  truncated  cones  joined  at  the  smallest  sections 
by  a  short  throat-piece.  At  A  and  B  there  are  air-chambers 
with  holes  for  the  insertion  of  piezometers,  by  which  the  fluid 


1 84 


HYDRA  ULICS. 


pressure  may  be  measured.     By  Art.  5,  Chap.  I,  the  theoretical 
quantity  Q  of  flow  through  the  throat  at  A  is 


at,  #a  being  the  sectional  areas  at  A  and  B,  respectively,  and 
fft  —  Hl  the  difference  of  head  in  the  piezometers,  or  the 
"head  on  Venturi,"  as  it  is  called. 


FIG.  116. 


Introducing  a  coefficient  of  discharge  C,  the  actual  delivery 
through  'A  is 


An  elaborate  series  of  experiments  by  Herschel  gave  C 
values  varying  between  .94  and  1-04,  but  the  great  majority  of 
the  values  lay  between  .96  and  .99. 

The  piezometers  may  be  connected  with  a  recorder,  and 
thus  a  continuous  register  of  the  quantity  of  water  passing 
through  the  meter  may  be  obtained  at  any  convenient  position 
within  a  radius  of  1000  ft.  This  distance  may  be  extended  to 
several  miles  by  means  of  an  electric  device. 

Other  meters  may  be  generally  classified  as  Piston  or  Re- 
ciprocating Meters  and  Inferential  or  Rotary  Meters.  They 
are  all  provided  with  recorders  which  register  the  delivery  with 
a  greater  or  less  degree  of  accuracy. 

The  piston  meter  (Fig.  1 1 8)  is  the  more  accurate  and  gives 
a  positive  measurement  of  the  actual  delivery  of  water  as 


METHODS   OF  GAUGING. 


185 


recorded  by  the  strokes  of  the  piston  in  a  cylinder  which  is 
filled  from  each  end  alternately.     Thus  an  additional  advan- 


l_     ..: 


PIG.    118. — SCHONHEYDER'S   POSITIVE 
METER. 


FIG.  119. — THE  UNIVERSAL 
METER. 


FIG.  120. — THE  BUFFALO  METER.       FIG.  121. — THE  UNION  ROTARY  PIS- 
TON METER. 

tage  possessed  by  a  water-engine  is  that  the  working  cylinder 
will  also  serve  as  a  meter. 

In  inferential  meters,  a  drum  or  turbine  is  actuated  by  the 
force  of  the  current  passing  through  the  pipe,  but  it  often 
happens  that  when  the  flow  is  small  the  force  is  insufficient  to 
cause  the  turbine  to  revolve,  and  there  is  consequently  no 
register  of  the  corresponding  quantity  of  water  passing  through 
the  meter. 


CHAPTER  VI. 


IMPACT. 

Note. — The  following  symbols  are  used  : 
z/,  =  the  velocity  of  the  jet  before  impact ; 
z>2  =    "         "         "     "     "   after  leaving  the  vane ; 
u  —    "          "         "     "    vane ; 

V  —    "  "     "    water  relatively  to  the  vane ; 

A  =  sectional  area  of  the  impinging  jet ; 
m  =  mass  of  the  water  reaching  the  vane  per  second. 
i.  Impact  of  a  Jet  upon  a  Flat  Vane  oblique  to  the 
Direction  of  the  Jet.— Let  6  be  the  angle  between  the  nor- 
mal to  the  vane  and  the  direction  of  the  impinging  jet,  <p  the 

angle  between  the  nor- 
mal to  vane  and  the 
direction  of  the  vane's 
motion,  and  a  the  an- 
gle between  the  vane 
and  the  vertical. 

The  jet  moving  with 
its  stream-lines  paral- 
lel, swells  out  near  the 
vane,  over  which  it 
spreads-and  with  which 
it  travels  along  in  the 
direction  of  the  vane's 
motion,  and  finally  again  flows  along  with  its  stream-lines  sen- 
sibly parallel  to  the  vane. 

The  problem  is  still  further  complicated  by  the  production 
of  eddies  and  vortices  for  which  allowance  can  only  be  made  in 
a  purely  empirical  manner. 

Let  N  be  the  normal  pressure  on  the  vane  due  to  the  impact. 
Let  N'  be  the  total  normal  pressure  on  the  vane. 
Let  W  be  the  weight  of  water  on  the  vane. 

1 86 


IMPACT.  IS/ 

Then 
N  =  N'  —  W  sin  a  =  change  of  momentum  in  direction  of  the 

normal 

=  mv^  cos  6  —  mu  cos  0. 
or 

N  =  m(vl  cos  6  —  u  cos  0).      i    .,     .     .     (i) 

(N.  B.  —  The  sign  in  front  of  u  cos  0  will  be  plus  if  the  jet 
and  vane  move  in  opposite  directions.) 

The  term  W  sin  a  may  be  designated  the  static  pressure 
and  the  term  m(vl  cos  6  —  u  cos  0)  the  dynamic  pressure 
which  causes  the  deviation  of  the  stream-lines. 

Note.  —  The  pressure  when  a  jet  first  strikes  the  plane  is 
greater  than  when  the  flow  has  become  steady,  or  permanent 
regime  is  established. 

This  is  made  evident  by  the  following  consideration  : 

At  any  moment  let  MN,  PQ,  RS  be  the  bounding  planes 
across  which  the  water  is  flowing  with  its  stream-lines  sensibly 
parallel. 

In  a  unit  of  time  let  the  bounding  planes  of  the  mass  be 
M'N',  P'Q,  R'S'. 

Then,  initially,  the  reaction  of  the  plane  must  destroy  the 
motion  of  the  mass  of  the  fluid  bounded  by  M'N',  PfQf, 
and  RfSf. 

Take  OC  to  represent  vl  in  direction  and  magnitude. 


In  one  second  the  vane  AB  moves  parallel  to  itself  into 
the  position  A'B'.     Let  A'  B'  intersect  OC  in  D. 
Then 

m  =  -A  .  DC  =  -A(v,  -  OD) 
g  g 

W    A(  COS  0\  f    N 

=  —A(vl—  u  --  4-J  ........     (2) 

g     \  cos  Ql 

Thus  equation  i  becomes 

7JJ         A 

N=  ---  s(^i  cos  6  —  u  cos  0)a.      •••     (3) 

g    COS  C7 


1  88  HYDRAULICS. 

Let   P  be   the    pressure   in   the   direction   of  the  vane's 
motion,  then 


,   .     (4) 


and  the  useful  work  done  on  the  vane  per  second 

=  Pu  =  —  A  -  ^u(v  cos  6  —  u  cos  0)2.  (c) 

g       cos  6 

<2jn         <7j 

The  total  available  work  =  —  A%-  ........     (6) 

g      2  ^ 


W      .  COS  0    , 

~^A  ^rtu(v*  cose  -  u  cos  0)a 


Hence,  the  efficiency  = 


cose-ucos^    (7) 
This  is  a  maximum  when 

z/t  cos  6  =  $u  cos  0, (8) 

and  therefore 

o 

the  maximum  efficiency  =  —  cos9  0.     .    .     .     (9) 

If  the  vane  is  of  small  sectional  area  a  portion  of  the  water 
will  escape  over  the  boundary  and  the  pressure  must  neces- 
sarily be  less  than  that  given  by  equation  3. 

Instead  of  one  vane  moving  before  the  jet,  let  a  series  of 
vanes  be  introduced  at  short  intervals  at  the  same  point  in  the 
path  of  the  jet. 

The  quantity  of  water  now  reaching  the  vane  per  second  is 
evidently 

m  =  -Avl9      .......     (10) 

o 


IMPACT.  189 

and,  by  equation  I,  the  normal  pressure 

fl     =  ^N—Avfa  costi  —  u  cos  0).      .     ./   .     (11) 

o 

Also,   the    pressure   in    the   direction   of    the    motion    of 
the  vane 

=  P  =  N  cos  0  =  -  —  A  cos  0  v1(vl  cos  0  —  u  cos  0).    (12) 

o 

The  useful  work  done  per  second 


—  Pu=  ^A  cos  0  ^X^i  cos  6  —  u  cos  0),    .     .     (13) 

o 


and  the  efficiency 


IV 

—  A  cos  0  v^u(vl  cos  6  —  u  cos  0) 


2  COS  0  «(Z/    COS  6  —  U  COS  0) 

~~ 


This  is  a  maximum  when  vl  cos  8  =  2u  cos  0,      .     .     (15) 
and  therefore 

the  maximum  efficiency  =  ---  .....     (16) 


SPECIAL  CASE  I.     Let  a  single  vane  be  at  right  angles  to, 
and  move  in  the  line  of,  the  jet's  motion,  Fig.  123. 
Then  6  =  o  =  0. 
Hence 

the  pressure  =  P=  N  ==  —  A(VI  —  u)9;      .     (17) 

"'     ~~      -r-     — 


FIG.  123.         the  useful  work  =  Pu  =  — Au(v^  —  #)3;    .     (18) 

o 


19°  HYDRA  ULICS. 


the  efficiency  =  2—(vl  —  u) ;      .     .    (19) 


o 

the  maximum  efficiency^  — (20) 

Again,  if  u  =  o,  i.e.,  if  the  vane  be  fixed,  and  if  H  be  the 
head  corresponding  to  the  velocity  vlt  then,  by  equation  17, 


P  =     Av?  =  2wAH 

=  twice  the  weight  of  a  column  of  water 

of  height  H  and  sectional  area  A. 

SPECIAL  CASE  2.     Let  each  of  a  series  of  vanes  be  at  right 
angles  to  and  move  in  the  line   of  the  jet's   motion   at   the 
instant  of  impact. 
Then  6  =  o  =  0. 


w     $ 
The  pressure  =  N  =  P  =  —  Av\(v^  —  u).       .     (21) 

<5 


IV 

The  useful  work  =  Pu     =  — A^lu(i'1  —  u^     .     (22) 

o 


The  efficiency  =  2*fo  "  *>.       .     .     (23) 


The  maximum  efficiency=  - (24) 


2.  Reaction — Jet  Propeller. — The  term  reaction  is  em- 
ployed to  denote  the  pressure  upon  a  surface  due  to  the  di- 
rection and  velocity  with  which  the  water  leaves  the  surface. 
Water,  for  example,  issues  under  the  head  h  and  with  the 


IMPACT.  IQI 

velocity  v  (at  contracted  section)  from  an  orifice  of  sectional 
area  A  in  the  vertical  side  of  a  vessel, 
Fig.  124. 

Let  R  be  the  reaction  on  the  opposite 
vertical  side  of  the  vessel,  and  let  Q  be 
the  quantity  of  water  which  flows  through 
the  orifice  per  second.  Then 

R  =  horizontal  change  of  momentum 

wQ         w 
= v  —  — CcAv*  —  2wcccvAh  —  2wAh,    .     .     .     (i) 

o  e» 

disregarding  the  contraction  and  putting  cv  —  I. 

Thus  the  reaction  is  double  the  corresponding  pressure 
when  the  orifice  is  closed  (Special  Case  I,  Art.  i). 

Again,  let  the  vessel  be  propelled  in  the  opposite  direction 
with  a  velocity  u  relatively  to  the  earth. 

Then  vl  —  u  is  the  velocity  of  the  jet  at  the  contracted 
section  relatively  to  the  earth  and 

R  =  horizontal  change  of  momentum 

=  ^Q(Vl-u) .     .     (2) 

o 

The  useful  work  done  by  the  jet 

IV 

=  Ru  =  —Qu(vl-u) (3) 

o 

The  energy  carried  away  by  the  issuing  water 


Hence 


w                          w    (v.  —  uY 
the  total  energy  =  —Qu(v,  -u)  +  —Q — 


(5) 


IQ2  HYDRAULICS.         , 

and 

w 

g  2U 

the  efficiency  =  —    —5 r  = . — .    .     .     .     (6) 

w     v,    —  u        v,  -4-  u 


Thus  the  more  nearly  vl  is  equal  to  u,  and  therefore  the 
larger  the  area  A  of  the  orifice,  the  greater  is  the  efficiency. 

If  the  vessel  is  driven  in  the  same  direction  as  the  jet,  then 
77,  -f-  u  is  the  relative  velocity  of  the  jet  with  respect  to  the 
earth,  and  the  reaction  is 

R  —  horizontal  change  of  momentum 


-G&  +  u)  =        c^Av^v,  +  u) 


(7) 


disregarding  the  contraction  and  putting  c,  =  I. 

3.  A  Jet  of  Water  impinging  upon  a  Surface  of  Rev- 
olution moving  in  the  Direction  of  its  Axis  and  also  in 
the  Line  of  the  Jet's  Motion.  —  Disregarding  friction,  the 
water  flows  over  the  surface  without  any  change  in  the  magni- 
tude of  the  relative  velocity  vt  —  u,  but  the  stream-lines  are 
deviated  from  their  original  direction  through  an  angle  /?. 

(N.B.  —  The  sign  before  u  is  plus  if  the  surface  and  jet  are 
moving  in  opposite  directions.) 

Let  the  water  leave  the  surface  at  D,  and  in  the  direction 
of  the  tangent  at  D  take  DE  to  represent  vl  —  u  in  direction 
and  magnitude.  Also  draw  DF  parallel  to  the  axis  of  the  sur- 
face and  take  DF  to  represent  //, 

Complete  the  parallelogram  EF. 

The  diagonal  DG  evidently  represents  in  direction  and 
magnitude  the  absolute  velocity  v^  with  which  the  water  leaves 


IMPACT.  193 

the  surface.     Hence,  from  the  triangle  DFG,  since  the  angle 
DFG  =  n  —  ft, 

v*  =  (vl  —  uj  +  u*  -  2(z/1  —  *)«  cos  (?r  —  /?), 
from  which 

/? 

z/j3  —  v*  =  2^(^j  —  w)(i  —  cos  /?)  =  4«(z/  —  u)  sin2  — .     (i) 

Also,  — -A(vl  —  u)  =  the  quantity   of  water  reaching  the 

<b 
surface  per  second. 

Hence,  if  P  is  the  pressure  in  the  direction  of  motion,  the 
useful  work  done  per  second 


FIG.  125 

-1  =  2—Au(vl  —  u)*  sina  — ;       .     (2) 

and  the  pressure 

0"  2  \«J/ 

The  efficiency 


A     f  \a     -    •>. 

—  Au(vl  —  u)  sm  — 

sm^.   .     (4) 


W    -V  v*  2 

A 


194  HYDRAULICS. 

This  is  a  maximum  when 

»,  =  3«,      •    •  >   -    •"  ...    (5) 
and  therefore 

the  maximum  efficiency  =  —  sin2  -.    .     .     .     (6) 

If,  instead  of  one  surface,  a  series  of  surfaces  are  succes- 
sively introduced  at  short  intervals  at  the  same  point  in  the 
path  of  the  jet,  the  quantity  of  water  reaching  each  surface 
per  second  becomes 

w 

m=       " (7) 


and  hence  the  useful  work,  pressure,  and  efficiency  also  respec- 
tively become 

w                               ft 
2~A^u(^-u)sm9~', (8) 


—Avfa  —  w)sina— ; (9) 

u(vl  —  u}    .  2,/? 


4"       V*  2 


The  efficiency  is  a  maximum  when 

v. 


(ii) 


Q 

its  value  then  being  sina  — . 

2 


It  will  be  observed  that  the  results  given  by  equations  2 
to  II  are  identical  with  those  given  by  equations  17  to  20  and 
21  to  24,  Art.  I,  except  that  in  each  case  there  is  an  additional 

ft 
factor  2  sin8       or  I  —  cos  ft.    This  factor  is  greater  than  unity, 

and  therefore  the  pressure,  useful  work,  and  efficiency  are  each 


IMPACT. 


195 


increased,  if  ft  >  90°,  i.e.,  in  the  case  of  a  concave  vane ;  while 
in  the  case  of  a  convex  vane,  ft  being  <  90°,  the  factor  is  also 
less  than  unity  and  they  are  each  diminished. 

SPECIAL  CASE.  Let  fi  =  180°,  i.e.,  let  the  vane  be  of  the 
cup  type  and  in  the  form  of  a  hemisphere. 

1 80° 

The  maximum  efficiency  is  sin"  =  unity,  and  is  per- 
fect. The  water  should  therefore  leave  the  surface  without 
velocity;  and  this  is  the  case ;  for,  by  equation  I, 


Hence 


v*  —  v*  =  ^ii(v^  —  u),     and     u  =  — . 

2 


v*  —  v*  =  v*,      and  therefore     ^a  =  o. 


4.  Impact  of  a  Jet  of  Water  upon  a  Vane  with  Borders. 
—  Let  the  vane  in  Art.  i  be  provided  with  borders,  Figs. 
126,  127,  so  as  to  produce  a  further  deviation  of  the  stream- 
lines, and  let  the  water  finally  flow  off  with  a  velocity  v*  in  a 
•direction  making  an  angle  0'  with  the  normal  to  the  vane. 


FIG.  126. 


FIG.  127. 


Then 
the  normal  pressure  =  N 

=  mvt  cos  0  T  mv^  cos  tf  3=  mu  cos  0 
=  m(vl  cos  0  ^F  z>a  cos  0'  =F  u  cos  0), 

the  sign  of  the  second  term  being  plus  or  minus  according  to 
the  direction   in  which  the  stream-lines  are  finally  deviated. 


196  HYDRAULICS. 

The  effect  of  the  borders  is  therefore  to  increase  or  diminish 
the  normal  pressure,  and  hence  also  the  useful  work  and  the 
efficiency. 

SPECIAL  CASE.  Let  the  vane  be  at  rest,  i.e.,  let  u  =  o,  and 
let  the  final  and  initial  directions  of  the  jet  be  parallel. 

Also,  let  v1  =  Vf     Then 

N  =  m(v^  cos  6  -\-  vl  cos  6) 

w 
=  2—Av?  cos  6 

o 

=  4wAH  cos  6. 

Hence,  if  fl—  o,  the  normal  pressure  N=  qwAH  •=•  four 
times  the  weight  of  a  column  of  water  of  height  H  and  sec- 
tional area  A. 

5.  Pressure  of  a  Steady  Stream  in  a  Uniform  Pipe 
against  a  Thin  Plate  AB  Normal  to  the  Direction  of 
Motion.  —  The  stream-lines  in  front  of  the  plate  are  deviated 
and  a  contraction  is  formed  at  Cf^  They  then  converge, 
leaving  a  mass  of  eddies  behind  the  plate. 

Consider  the  mass  bounded  by  the  transverse  planes  ClCl> 
3  ,  where  the  stream-lines  are  again  parallel. 
At  C£i  let  A  ,  Al  ,  vl  ,  zl  be  the  mean  intensity  of  the  press- 

ure, the  sectional  area  of  the 
waterway,  the  velocity  of  flow,  and 
the  elevation  of  the  C.  of  G.  of 
the  section  above  datum. 

Let  /2,   AS,   z>3,    z^   be  corre- 
sponding symbols  at  Cfv 

Let  /3>   A19   vlt  *„  be   corre- 
sponding symbols  at  C9C3. 

Let  a  be  the  area  of  the  plate. 

Let  cc  be  the  coefficient  of  contraction. 

Neglect  the  skin  and  fluid  friction  between  ClCl  and 

Then  by  Bernoulli's  theorem, 


+  + 

'  '  '  ' 


W  2g  W          2g  W  2g  2g 


IMPACT.  197 

(v  —  v\ 
the  term  —  -  —  representing  the  loss  of   head  due  to  the 

bending  of  the  stream-lines  between  Cf^  and  C3C3. 
Hence 

•  A  -A      (v*  -  v>Y 


Again,  let  R  be  the  total  pressure  on  the  plane.     Then 

x  .  x  A         (  fluid  pressure   in  the   direction 

A  -M,  =  (A  -  AK  =  |     of  the  axis_ 


—  2* 


=  component  of  the  weight  in  the  direction 

of  the  axis. 
Thus 
^  __  j>s)Al  +  wAl(zl  —  ^,)  —  R  =  change  of  motion  in  direction 

of  axis 
=  0, 

since  the  motion  is  steady. 
Hence 

R       A -A (",-".)' 


wA  l  w  2g 

But  A^,  =  AM  =  cc(Al  —  a)Vr     Therefore 


=-*${&&>- >} 

v?      (         m  }  a 

=  wa  —  m  \  —, r  —  I  [•  , 

2g     \  cc(m  -  i)          j 


A 

where  m  =  — ,  or 
a 


R  = 


r         m  ) 

where  K  —  in  \  —, —      — r  —  I  > 

\  cc(m  -  i)          f 


198  HYDRAULICS'. 

6.  Pressure  of  a  Steady  Stream  in  a  Uniform  Pipe  on 
a  Cylindrical  Body  about  Three  Diameters  in  Length.— 

The  stream-lines  in  front  of  the  body  are  deviated  and  a 
contraction  is  formed  at  C9Ct.  They  then  converge,  flow  in 
parallel  lines,  and  converge  a  second  time  at  C3C3,  leaving  a 
mass  of  eddies  behind  the  body. 

Consider  the  mass  bounded  by  the  planes  C£^  CtC4. 
As  in  the  previous  article,  let 

/>,,  Alt  vl,  zl  be  the  intensity  of  pressure,  sectional  area  of 
the  waterway,  velocity  of  flow,  and  elevation 
of  C.  of  G.  above  datum  at 


r  A,  ^2,  z>2,  <sra  be  similar  symbols 

V3          <?4  <• 

for 


/3,  ^48,  ^3,  #3  be   similar  symbols 

for  Cfy 
&    &*  3-  p^Al,vl,Zi  be   similar  symbols 

FIG-  I29'  for  C&. 

Neglect  the  skin  and  fluid  friction  between  ClCl  and  C4Ct. 
Then,  by  Bernouilli's  theorem, 


,  . 

W       2g  W        2g  w    '    2g 

,  A   ,  *>?  .  {^s  -  *$   , 

!*4++"t  " 


^""«    being  the  loss  of  head  between  <7a£,  and  C,C3  and 
—  being  the  loss  of  head  between  C3C9  and  CtCt. 


o 

Hence 


*    i  A—  A  _  (^.  —  ^)a  I  (^  -  Oa 

J"  4-f  ~^^     "IF      IF" 


But  A&  =  ^3e;a  =  ^3^,, 

and  A3  =  A,  —  a. 


IMP  A  CT.  199 

Therefore 


,        y      t      A_       _A^n 

a       I        \7JJ^-a)      A,-  a]  J 


where  m  =  — *. 

Also,  as  in  the  preceding  article, 

(A- 

Hence 


f 
2g       (m—  i)2      (m  -  i)a  V, 


where  m  =  —  -,  and 
a 


This  value  of  K  is  always  less  than  the  value  of  K  for  the 
plate  in  the  preceding  article  for  the  same  values  of  m,  a, 
and  cf 

Hence  the  pressure  on  the  cylinder  is  also  less  than  the 
corresponding  pressure  on  the  plate. 

In  every  case  K  should  be  determined  by  experiment. 

7.  Jet  impinging  upon  a  Curved  Vane  and  deviated 
wholly  in  one  Direction  —  Best  Form  of  Vane.  —  Let  the 
jet,  of  sectional  area  A,  moving  in  the  direction  AB  with  a 
velocity  v^  ,  drive  the  vane  AD  in  the  direction  AC  with  a 
velocity  u. 


200 


HYDRAULICS. 


Take  AB  to  represent  v^  in  direction  and  magnitude. 
"      AC  "          "         u   "         "          ". 

Join  CB. 

Then  CB  evidently  represents  F,  the  velocity  of  the  water 
relatively  to  the  vane,  in  direction  and  magnitude.  If  CB  is 
parallel  to  the  tangent  to  the  vane  at  A,  there  will  be  no  sud- 


FIG.  130. 

den  change  in  the  direction  of  the  water  as  it  strikes  the  vane, 
and,  disregarding  friction,  the  water  will  flow  along  the  vane 
from  A  to  D  without  any  change  in  the  magnitude  of  the  rela- 
tive velocity  V  (=  CB).  The  vane  is  then  said  to  "receive  the 
water  without  shock."* 

Again,  from  the  triangle  ABC,  denoting  the  angles  BA  C, 
ABC,  ACB,  byA,£,  C,  respectively. 


sin  B 


u  _  AC  _  sin  B  __ 

^  ="  ~AB  ~  sin  C  ~  sin  (A  +  B)'  '     ' 


.     .     (I) 


and  therefore 


cot  B  =  —  cosec  A  —  cot  A,      ....     (2) 


IMPACT.  201 

a  formula  giving  the  angle  between  the  lip  and  the  direction 
of  the  impinging  jet,  which  will  ensure  the  water  being  received 
"  without  shock." 

In  the  direction  of  the  tangent  to  the  vane  at  D,  take 
DE  =  CB  (=  V). 

Draw  DF  parallel  and  equal  to  AC(=  u). 

Complete  the  parallelogram  EF. 

Then  the  diagonal  DG  evidently  represents  in  direction  and 
magnitude  the  absolute  velocity  v^  with  which  the  water  leaves 
the  vane. 

Draw  AK  equal  and  parallel  to  DG  (=  z/a). 

Join  BK.  Then  BK  represents  the  total  change  of  velocity 
between  A  and  D  in  direction  and  magnitude. 

Thus,  if  R  is  the  resultant  pressure  on  the  vane,  then 
R  =  m.  BK. 

Let  ML  be  the  projection  of  BK  upon  AC. 

Then  ML  represents  the  total  change  of  velocity  in  the 
direction  of  the  vane's  motion. 

Let  P  be  the  pressure  upon  the  vane  in  this  direction. 

Then 

P=m.  LM. (3) 

The  useful  work  =  Pu  =  mu  .  LM  =  mV*    ~  V* .      .     .     (4) 


w  A  v? 


The  total  available  work  =  -  A  -- (5) 


„,,        ~  .          mu.  LM  v?  —  v* 

The  efficiency  --  —  =  img—  --  r-  ......     (6) 

w  Av?  *  wAv? 


Again,  join  CK.- 

Then,  since  A  C  is  equal  and  parallel  to  DF,  and  AK  to  DG, 
the  line  CK  is  equal  and  parallel  to  DE,  and  is  therefore  equal 
to  CB. 

Thus  in  the  isosceles  triangle  CBK,  CB  is  equal  and  parallel 
to  the  relative  velocity  Fat  A,  CK  is  equal  and  parallel  to  the 


2O2  HYDRA  ULICS. 

relative  velocity  Fat  D,  and  the  base  B  K  represents  the  total 
change  of  motion. 

Let  8  be  the  angle  through  which  the  direction  of  the  water 
is  deviated,  i.e.,  the  angle  between  AB  and  AK.     Then 


=  V*  -\-  U*  —  2V  Ji  COS  (A  +  #),          ......      (7) 

and  also 

F3  =  CK*  =  CB*  =  AB*  +  AC*  -  2AB  .  AC  cos  A 
=  v*  -\-u*  —  2v  ji  cos  A  ..........     (8) 

Hence 


— L  =  u  \  vt  cos  (A  +  6)  —  vl  cos  A } .     .     .     (9) 


If  BH  is  drawn  parallel  to  the  tangent  at  D,  BK  evidently 
bisects  the  angle  between  BC  and  BH,  and  this  angle  is  equal 
to  the  angle  between  the  tangents  to  the  vane  at  A  and  D. 

Let  a  be  the  sttpplcmcnt-^f  the  angle  between  the  normals 
at  A  and  D.  Then  the  angle  KCB  —  a,  and 


the  angle  CBK  =  -(180°  -  «)  =  90°  -  £ 

2  2 


Therefore 

BK  =  2CB  (cos  00°  —  -  ]  =  2Fsin  -. 
\  21  2 

Hence 


;in- (10) 


IMP  A  CT.  2O3 

Let  X,  Fbe  the  components  of  R  in  the  direction  of  the 
normal  at  A  and  at  right  angles  to  this  direction.     Then 

Y=R  cos-  =  mVsm  or;     ....     (n) 


X  =  R  sin  —  =  2m  V  sin3  -  =  m  V(  i  —  cos  a).          (  1  2} 

2  2 


The  efficiency  is  a  maximum  when 

dP 


The  efficiency  is  nil  when 

Pu  =  o,     i.e.,  when  u  =  o  or  P  =  o.     .     .     .     (14) 

In  the  latter  case,  since  P  —  m.  LM,  the  projection  LM 
must  be  nil,  and  therefore  BK  must  be  at  right  angles  to  A  C, 
as  in  Fig.  131. 

FIG.  131. 


FIG.  132. 


204  HYDRA  ULICS. 

The  angle  ACB  is  now  =  180°  --  ,  and  therefore 

u_       sin  ABC 

vl  ~~  sin  A  CB 


sn 


in  (180°  —  -^ 

(IS) 


sm  — 

2 


If  BK  is  parallel  to  AC  (Fig.  132),  then 

the  angle  ACB  =  -(180°  -«)  +  «  =  90°  +  - 

2  2 

.and  therefore 

„      sin  (90°  +  -  +  A\      cost-  ~4- A] 
u_  _  sin  ABC  V       r  2          )  _         \2          1 


sm  I  Qcr  +  - 1  cos  - 

SPECIAL  CASE. — Let  the  direction  of  the  impinging  jet  be 
tangential  to  the  vane  at  A,  and  let  the  jet  and  vane  move  in 
the  same  direction.  Then 

V—  v.  —  uy      m  =  — A(v.  —  11) ; 
g 

P  =  Y=  — A(vt  —  u)\i  —  cos  a)  =  2 — A(v^  —  u)  sin2  -; 

«5  o 

W  & 

useful  work  =  Pu  =  2 — Ati(v,  —  uY  sin2  —  ; 
g  2 

U(V.   —  U}"  OL 

efficiency  =  4  —  sin  — . 


IMP  A  CT.  20$ 

This  is  a  maximum  and  equal  to  —  sin2  —  when  vl  =  $u. 

27  2 

These  results  are  identical  with  those  for  a  concave  cup 
when  a  =  180°. 

Instead  of  one  vane  let  a  series  of  vanes  be  successively 
introduced  at  short  intervals  at  the  same  point  in  the  path  of 
the  jet.  Then 

w 
m  =  —Av^ 

and  hence  the  pressure  P,  useful  work,  and  efficiency  respec- 
tively become 

—A 

o 


w  A 
—  Av,  . 

S 


and 


8.  Friction. — The  effect  of  friction  has  been  disregarded, 
and  nothing  definite  is  known  as  to  its  action  or  law  of  distri- 
bution.    It  has  been  suggested  to  assume  that  the  loss  of  head 
due  to  friction  is  a  fraction  of  the  head  due  to  the  velocity  of 
the  jet  relatively  to  the  surface  over  which  it  spreads.     Thus 
in  Art.  7 

V* 
the  loss  of  head  due  to  friction  =/ — 

V* 
and  the  corresponding  loss  of  energy  =  wQ*f — • 

9.  Resistance  to  the  Motion  of  Solids  in  a  Fluid  Mass. 

— The    preceding   results   indicate    that    the    pressure  due  to 


2O6  HYDRA  ULICS. 

the  impact  of  a  jet  upon  a  surface  may  be  expressed  in  the 
form 


A  being  the  sectional  area  of  the  jet,  V  the  velocity  of  the  jet 
relatively  to  the  surface,  and  K  a  coefficient  depending  on  the 
position  and  form  of  the  surface. 

Again,  the  normal  pressure  (N)  on  each  side  of  a  thin 
plate,  completely  submerged  in  an  indefinitely  large  mass  of 
still  water,  is  the  same.  If  the  plate  is  made  to  move  hori- 
zontally with  a  velocity  F,  a  forward  momentum  is  developed 
in  the  water  immediately  in  front  of  the  plate,  while  the  plate 
tends  to  leave  behind  the  water  at  the  back.  A  portion  of  the 
water  carried  on  by  the  plate  escapes  laterally  at  the  edges 
and  is  absorbed  in  the  neighboring  mass,  while  the  region  it 
originally  occupied  is  filled  up  with  other  particles  of  water. 
Thus  the  normal  pressure  N,  in  front  of  the  plate,  is  increased 
by  an  amount  n,  while  at  the  back  eddies  and  vortices  are  pro- 
duced, and  the  normal  pressure  N  at  the  back  is  diminished 
by  an  amount  n'  .  The  total  resultant  normal  pressure,  or  the 
normal  resistance  to  motion,  is  n-\-  n',  and  this  increases  with 
the  speed.  In  fact,  as  the  speed  increases,  n'  approximates 
more  and  more  closely  to  N,  and  in  the  limit  the  pressure 
at  the  back  would  be  nil,  so  that  a  vacuum  might  be  main- 
tained. 

Confining  the  attention  to  a  plate  moving  in  a  direction 
normal  to  its  surface,  the  resistance  is  of  the  same  character  as 
if  the  plate  is  imagined  to  be  at  rest  and  the  fluid  moving 
in  the  opposite  direction  with  a  velocity  V.  So,  if  both  the 
water  and  the  plate  are  in  motion,  imagine  that  a  velocity 
equal  and  opposite  to  that  of  the  water  is  impressed  upon 
every  particle  of  the  plate  and  of  the  water.  The  resistance  is 
then  of  the  same  character  as  that  of  a  plate  rrioving  in  still 
water,  the  velocity  of  the  plate  being  the  velocity  relatively 
to  the  water.  Thus,  in  general,  the  resistance  to  the  motion 
of  such  a  plane  moving  in  the  direction  of  the  normal  to  its 


IMPACT.  207 

surface,  with  a  velocity  V  relatively  to  the  water,  may  be  ex- 
pressed in  the  form 


R  -  KwA  -  , 


A  being  the  area  of  the  plate,  and  K  a  coefficient  depending 
upon  the  form  of  the  plate  and  also  upon  the  relative  sectional 
areas  of  the  plate  and  of  the  water  in  which  it  is  submerged. 

According  to  the  experiments  of  Dubuat,  Morin,  Piobert, 
Didion,  Mariotte,  and  Thibault,  the  value  of  K  may  be  taken 
at  1.3  for  a  plate  moving  in  still  water,  and  at  1.8  for  a  current 
moving  on  a  fixed  plate.  Unwin  points  out  the  unlikelihood 
of  such  a  difference  between  the  two  values,  and  suggests  that 
it  might  possibly  be  due  to  errors  of  measurement. 

Again,  reasoning  from  analogy,  the  resistance  to  the  motion 
of  a  solid  body  in  a  mass  of  water,  whether  the  body  is  wholly 
or  only  partially  immersed,  has  been  expressed  by  the 
formula 


R  =  KwA—, 

V  being  the  relative  velocity  of  the  body  and  water,  A  the 
greatest  sectional  area  of  the  immersed  portion  of  the  body  at 
right  angles  to  the  direction  -of  motion,  and  K  a  coefficient  de- 
pending upon  the  form  of  the  body,  its  position,  the  relative 
sectional  areas  of  the  body  and  of  the  mass  of  water  in  which 
it  is  immersed,  and  also  upon  the  surface  wave-motion. 
The  following  values  have  been  given  for  K\ 

K  =  i.i  for  a  prism  with  plane  ends  and  a  length  from  3  to  6 
times  the  least  transverse  dimension  ; 

K  =  i.o  for  a  prism,  plane  .in  front,  but  tapering  towards  the 
stern,  the  curvature  of  the  surface  changing  gradu^ 
ally  so  that  the  stream-lines  can  flow  past  without 
any  production  of  eddy  motion,  etc.; 


208  HYDRA  ULICS. 

K  —    .5  for  a  prism  with  tapering  stern  and  a  cut-water  or 

semi-circular  prow ; 
K  =  .33  for  a  prism  with  a  tapering  stern  and  a  prow  with  a 

plane  front  inclined  at  30°  to  the  horizon ; 
K  =  .16  for  a  well-formed  ship. 

Froude's  experiments,  however,  show  that  the  resistance  to 
the  motion  of  a  ship,  or  of  a  body  tapering  in  front  and  in 
the  rear,  so  that  there  is  no  abrupt  change  of  curvature  lead- 
ing to  the  production  of  an  eddy  motion,  is  almost  entirely 
due  to  skin-friction  (see  Art.  i,  Chap.  II). 


IMPACT.  209 


EXAMPLES, 

1.  A  stream  with  a  transverse  section  of  24  square  inches  delivers       y 
10  cubic  feet  of  water  per  second  against  a  flat  vane  in  a  normal  direc-  ^ 
tion.     Find  the  pressure  on  the  vane.  Am.  1171!  Ibs. 

2.  If  the  vane  in  question  i  moves  in  the  same  direction  as  the  im-    ./ 
pinging  jet  with  a  velocity  of  24  ft.  per  second,  find  (a)  the  pressure  on 

the  vane ;  (b)  the  useful  work  done ;  (c)  the  efficiency. 

Am.  (a)  4211  Ibs.;  (ff)  10,125  ft.-lbs.;  (c)  .288. 

3.  What  must  be  the  speed  of  the  vane  in  question  2,  so  that  the        J 
efficiency  of  the  arrangement  may  be  a  maximum  ?    Find  the  maximum     ^ 
efficiency.  Ans.  20  ft.  per  sec.;  ^V          % 

4.  Find  (a)  the  pressure,  (b)  the  useful  work  done,  (c)  the  efficiency, 
when,  instead  of  the  single  vane  in  question  2,  a  series  of  vanes  are  intro- 
duced at  the  same  point  in  the  path  of  the  jet  at  short  intervals. 

Ans.  (a)  703^  Ibs.;  (b}  16,875  ft.-lbs.;  (c)  .48. 

What  must  be  the  speed  of  the  vane  to  give  a  maximum  efficiency  ? 
What  will  be  the  maximum  efficiency?  Ans.  30  ft.  per  sec.;  .5. 

5.  A  stream  of  water  delivers  7,500  gallons  per  minute  at  a  velocity  of 
15  ft.  per  second  and  strikes  an  indefinite  plane.     Find  the  normal  pres- 
sure on  the  vane  when  the  stream    strikes  the  vane  (a)  normally;  (d)  at 
an  angle  of  60°  to  the  normal.  Ans.  (a)  585.9  Ibs.;  292.9  Ibs. 

6.  A  railway  truck,  full  of  water,  moving  at  the  rate  of  10  miles  an 
hour,  is  retarded  by  a  jet  flowing  freely  from  an  orifice  2  in.  square  in 
the  front,  2  ft.  below  the  surface.     Find  the  retarding  force. 

Ans.  7.97  Ibs. 

7.  A  jet  of  water  of  48  sq.  in.  sectional  area  delivers  100  gallons  per   Q% 

second  against  an  indefinite  plane  inclined  at  30°  to  the  direction  of  the- ( 

jet ;  find  the  total  pressure  on  the  plane,  neglecting  friction.     How  will 

the  result  be  affected  by  friction  ?  Ans.  750  Ibs.    ' 

8.  If  the  plane  in  question  7  move  at  the  rate  of  24  ft.  per  second  in 
a  direction   inclined  at  60°  to  the  normal  to  the  plane,  find  the  useful 
work  done  and  the  efficiency.  Ans.  2250  ft.-lbs.;  TV 

At  what  angle  should  the  jet  strike  the  plane  so  that  the  efficiency 
might  be  a  maximum?     Find  the  maximum  efficiency. 

Ans.  sin1  £ ;  -£.. 

9.  A  stream  of  32  square  inches  sectional  area  delivers  32  cub.  feet 
of  water  per  second.     At  short  intervals  a  series  of  flat  vanes  are  intro- 


210  HYDRA  ULICS. 

duced  at  the  same  point  in  the  path  of  the  stream.  At  the  instant  of 
impact  the  direction  of  the  jet  is  at  right  angles  to  the  vane,  and  the 
vane  itself  moves  in  a  direction  inclined  at  45°  to  the  normal  to  the 
vane.  Find  the  speed  of  the  vane  which  will  make  the  efficiency  a 
maximum.  Also  find  the  maximum  efficiency  and  the  useful  work 
done.  Ans.  15.08  ft.  per  sec.;  /T;  2io6f|-f  ft.-lbs. 

10.  In  a  railway  truck,  full  of  water,  an  opening  2  in.  in  diameter 
is  made  in  one  of  the  ends  of  the  truck,  9  ft.  below  the  surface  of  the 
water.     Find  the  reaction  (a)  when  the  truck  is  standing;  (b)  when  the 
truck  is  moving  at  the  rate  of  10  ft.  per  second  in  the  same  direction  as 
the  jet  ;  (c)  when  the  truck  is  moving  at  the  rate  of  10  ft.  per  second  in 
a  direction  opposite  to  that  of  the  jet.     If  this  movement  of  the  truck 
is  produced  by  the  reaction  of  the  jet,  find  the  efficiency. 

Ans.  (a)  24.55  Ibs.  per  sq.  in.;  (b)  34.78  Ibs.  per  sq.  in.;  (c)  14.3 
Ibs.  per  sq.  in.;  .588. 

11.  From  a  ship  moving  forward  at  6  miles  an  hour  a  jet  of  water  is 
sent  astern  with  a  velocity  relative  to  the  ship  of  30  feet  per  second  from 
a  nozzle  having  an  area  of  16  square  inches;  find  the  propelling  force 
and  the  efficiency  of  the  jet  as  a  propeller  without  reference  to  the  man- 
ner in  which  the  supply  of  water  may  be  obtained. 

Ans.   i 


12.  A  stream  of  64  sq.  in.  section  strikes  with  a  40-  ft.  velocity  against 
a  fixed  cone  having  an  angle  of  convergence  =  100°  ;  find  the  hydraulic 
pressure.  Ans.  492.1  Ibs. 

13.  A  jet  of  9  sq.  in.  sectional  area,  moving  at  the  rate  of  48  ft.  per 
second,  impinges  upon  the  convex  surface  of  a  paraboloid  in  the  direc- 
tion of  the  axis  and  drives  it  in  the  same  direction  at  the  rate  of  16  ft. 
per  second.     Find  the  force  in  the  direction  of  motion,  the  useful  work 
done,  and  the  efficiency.    The  base  of  the  paraboloid  is  2  ft.  in  diameter 
and  its  length  is  8  inches.  Ans.  25  Ibs.;  400  ft.-lbs.;  r£y. 

14.  A  stream  of  water  of  16  sq.  in.  sectional  area  delivers  12  cubic  feet 
of  water  per  second  against  a  vane  in  the  form  of  a  surface  of  revolu- 
tion, and  drives  in  the  same  direction,  which  is  that  of  the  axis  of  the 
vane.    The  water  is  turned  through  an  angle  of  120°  from  its  original 
direction   before   it   leaves  the  vane.       Neglecting    friction,   find   the 
speed  of  vane  which  will  give  a  maximum  effect.     Also  find  impulse 
on  vane,   the  work  on  vane,  and  the  velocity  with  which  the  water 
leaves  the  vane. 

Ans.  36  ft.  per  sec.;  562^  Ibs.;  20,250  ft.-lbs.;  95.24  ft.  per  sec. 

15.  At  8  knots  an   hour  the  resistance  of  the  Water-witch  was  5500 
Ibs.;  the  two  orifices  of  her  jet  propeller  were  each  18  in.  by  24  in. 
Find  (a)  the  velocity  of  efflux;  (b)  the  delivery  of  the  centrifugal  pump; 


IMPACT.  211 

(V)  the  useful  work  done ;  (d)  the  efficiency;  (<?)  the  propelling  H.P.,  as- 
suming the  efficiency  of  the  pump  and  engine  to  be  .4. 

Ans.  (a)  29.4  ft.  per  sec.;  (b)  1 104.6  gallons  per  sec.;  (c)  74,393  ft.- 
Ibs.;  (</).63;  (e)  532. 

1 6.  If  feathering-paddles    are  substituted   for  the  jet  propeller  in 
question  15,  what  would  be  the  area  of  stream  driven  back  for  a  slip  of 
25$  ?     Find  the  efficiency  and  the  water  acted  on  in  gallons  per  minute. 

Ans.  34  sq.ft.;  .75;  236,000. 

17.  A  vane  moves  in  the  direction  ABC  with  a  velocity  of  10  ft.  per 
second,  and  a  jet  of  water  impinges  upon  it  at  B  in  the  direction  BD 
with  a  velocity  of  20  ft.  per  second ;  the  angle  between  BC  and  BD  is 
30°.     Determine  the  direction  of  the  receiving-lip  of  the  vane,  so  that 
there  may  be  no  shock. 

Ans.  The  angle  between  lip  and  BC  =  23°47'. 

18.  A  jet  moves  in  a  direction  AltCwith  a  velocity  Fand  impinges 
upon  a  vane  which  it  drives  in  the  direction  BD  with  a  velocity  \  V. 
The  angle  ABD  is  165°.     Determine  the  direction  of  the  lip  of  the  vane 
at  B,  so  that  there  may  be  no  shock  at  entrance. 

Ans.  The  angle  between  lip  and  direction  of  stream  =  i4°3'. 

19.  A  jet  issues  through  a  thin-lipped  orifice  i  sq.  in.  in  sectional 
area  in  the  vertical  side  of  a  vessel  under  a  pressure  equivalent  to  a 
head  of  900  ft.  and  impinges  on  a  curved  vane,  driving  it  in  the  direc- 
tion of  the  axis  of  the  jet.     The  water  enters  without  shock  and  turns 
through  an  angle  of  60°  before  it  leaves  the  vane.     Find  (a)  the  speed 
of  the  vane  which  will  give  a  maximum  effect ;  (If)  the  pressure  on  the 
vane  ;  (c)  the  work  done  ;  (d)  the  absolute  velocity  with  which  the  water 
leaves  the  vane ;  (e)  the  reaction  on  the  vessel,  disregarding  contraction. 

Ans.  (a)  80  ft.  per  sec. ;  (d)  320.9  IDS.;  (c)  46.68  H.  P.;  (d}  184  ft. 
per  sec.;  (e)  781.25  Ibs. 

20.  A   stream    moving  with  a  velocity  v  impinges  without   shock 
upon  a  curved  vane  and  drives  it  in  a  direction  inclined  at  an  angle  to 
the  direction  of  the  stream.     The  angle  between  the  lip  of  the  vane  and 
the  direction  of  the  stream  is  x,  and  V  is  the  relative  velocity  of  the 
water  with  respect  to  the  vane.     If  the  speed  of  the  vane  is  changed  by 
a  small  amount,  say  n  per  cent,  show  that  the  corresponding  change  in 
the  direction  of  the  lip,  in  order  that  the  water  might  still  strike  the 

v 
vane  without  shock,  is  n —  sin  x. 

21.  A  jet  of  water  under  a  head  of  20  feet,  issuing  from  a  vertical 
thin-lipped  orifice  i  in.  in  diameter,  impinges  upon  the  centre  of  a  vane 
3  ft.  from  the  orifice.     Determine  the  position  of  the  vane  and  the  force 
of  the  impact  (a)  when  the  vane  is  a  plane  surface ;  (b)  when  the  vane  is 
6  in.  in  diameter  and  in  the  form  of  a  portion  of  a  sphere  of  6  in.  radius. 


2 1 2  HYDRA  ULICS. 

22.  A  stream  of  water  of  36  sq.  in.  section  moves  in  a  direction  ABC 
and  delivers  4  cub.  ft.  of  water  per  second  upon  a  vane  moving  in  a 
direction  BD  with  a  velocity  of  8  ft.  per  second,  the  angle  between  BC 
and  BD  being  30°.     Find  (a)  the  best  form  to  give  to  the  vane ;  (b)  the 
velocity  of  the  water  as  it  leaves  the  vane ;  (c)  the  mechanical  effect  of 
the  impinging  jet ;  and  (d)  the  efficiency,  the  angle  turned  through  by 
the  jet  being  90°. 

Ans.  (a)  The  angle  between  lip  and  BC  —  23°48';  (b)  2.946  ft, 
per  sec. ;  (c)  966.098  ft.  per  sec.;  (d)  .966. 

23.  A  stream  of  thickness   /  and    moving   with  the  velocity  v  im- 
pinges without  shock  upon  the  concave  surface  of  a  cylindrical  vane  of 
a  length  subtending  an  angle  20.  at  the  centre.     Determine  the  total 
pressure  upon  the  vane  (a)  if  it  is  fixed  ;  (b)  if  it  is  moving  in  the  same 
direction  as  the  stream  with  the  velocity  u.     In  case  (b)  also  find  (c)  the 
work  done  on  the  vane. 

iu  w  yu 

Ans.  (a)  2— bin*  sin  a;  (b)  2-bt(v  —  U)*  sin  a  ;  (c)  2—btu(y  —  u}*  sin5  a. 

O  £>  £ 

24.  Two  cubic  feet  of  water  are  discharged  per  second  under  a  press- 
ure of  loo  Ibs.  per  sq.  in.  through  a  thin-lipped  orifice  in  the  vertical 
side  of  a  vessel,  and  strike  against  a  vertical  plate.     Find  the  pressure 
on  the  plate  and  the  reaction  on  the  vessel.  Ans.  475.82  Ibs. 

25.  A  stream  moving  with  a  velocity  of  16  ft.  per  second  in  the  direc- 
tion ABC,  strikes  obliquely  against  a  flat  vane  and   drives  it  with  a 
velocity  of  8  ft.  per  second  in  the  direction  BD,  the  angle  CBD  being  30°. 
Find  {a)  the  angle  between  ABC  and  the  normal  to  the  plane  for  which 
the  efficiency  is  a  maximum  ;  (b)  the  maximum  efficiency ;  (c)  the  velocity 
with  which  the  water  leaves  the  vane;  (d}  the  useful  work  per  cubic 
foot  of  water. 

Ans.  (a)  21°  44';  (b)  .25664;  (c)  12.6  ft.  per  sec.;  (d)  256.64  ft.-lbs. 


CHAPTER  VII. 
HYDRAULIC   MOTORS   AND   CENTRIFUGAL   PUMPS. 

I.  Hydraulic  Motors  are  machines  designed  to  utilize  the 
•energy  possessed  by  a  moving  mass  of  water  in  virtue  of  its 
position,  pressure,  and  velocity. 

The  motors  may  be  classified  as  follows : 

(1)  Bucket  Engines. — In  this  now  antiquated  form  of  motor 
weights  are  raised  and  resistances  overcome  by  allowing  water 
to  flow  into  suspended  buckets,  thus  causing  them  to  descend 
vertically. 

(2)  Rams  and  Jet-pumps,  in  which  the  impulsive  effect  of 
one  mass  of  water  is  utilized  to  drive  a  second  mass  of  water. 

(3)  Water-pressure  Engines  are  especially  adapted  for  high 
pressures   and    low  speeds,  and    necessarily  have   very  heavy 
moving  parts.     With  low  pressures  the  engine  becomes  un- 
wieldy and  costly. 

Pressure-engines  are  either  reciprocal  or  rotative.  The 
latter  are  very  convenient  with  moderately  high  pressures  and 
-especially  when  they  are  to  drive  machinery  which  is  to  be  used 
intermittently.  They  also  give  an  exact  measurement  of  the 
water  used. 

Direct-acting  pressure-engines  are  of  great  advantage  where 
a  slow  and  steady  motion  is  required,  as,  for  example,  in  work- 
ing cranes,  lifts,  etc. 

(4)  Vertical  Wat  er-iv  heels,  in  which  the  water  acts  almost 
wholly  by  weight,  or  partly  by  weight  and  partly  by  impulse, 
or  wholly  by  impulse. 

(5)  Turbines,  in  which  the  water  acts  wholly  by  pressure 
or  wholly  by  impulse. 


214 


HYDRAULICS. 


2.  Hydraulic  Rams. — By  means  of  the  hydraulic  ram  a 
quantity  of  water  falling  through  a  vertical  distance  hl  is  made 
to  force  a  smaller  weight  of  water  to  a  higher  level. 

The  water  is  brought  from  a  reservoir  through  a  supply- 
pipe  5.  At  the  end  B  of  this  pipe  there  is  a  check-  or  clack- 
valve  opening  into  an  air-chamber  A,  which  is  connected  with 
a  discharge-pipe  D.  At  C  there  is  a  weighted  check-  or  clack- 
valve  opening  inwards,  and  the  length  of  its  stem  (or  the  stroke) 
is  regulated  by  means  of  a  nut  or  cottar  at  E.  When  the  waste- 
valve  at  C  is  open  the  water  begins  to  escape  with  a  velocity  due 
to  the  head  hl  and  suddenly  closes  the  valve.  The  momentum. 


FIG.  133. 

of  the  water  in  the  pipe  opens  the  valve  at  B,  and  a  portion  of 
the  water  is  discharged  into  the  air-vessel.  From  this  vessel  it 
passes  into  the  discharge-pipe  in  consequence  of  the  reaction 
of  the  compressed  air.  At  the  end  of  a  very  short  interval  of 
time  the  momentum  of  the  water  has  been  destroyed,  the  valve 
at  B  closes,  the  waste-valve  again  opens,  and  the  action  com- 
mences as  before.  It  is  found  that  the  efficiency  of  the  ram  is 
increased  by  introducing  a  small  air-vessel  at  F,  supplied  with 
a  check-  or  clack-valve  opening  inwards  at  G.  The  wave-motion 
started  up  in  the  supply-pipe  by  the  opening  and  closing  of  the 
valve  at  B  has  been  utilized  in  driving  a  piston  so  as  to  pump 
up  water  from  some  independent  source. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS.    21$ 

Let  v  be  the  velocity  of  flow  in  the  supply-pipe  at  the  mo- 
ment when  the  valve  at  C  is  closed. 
"     Wl  be  the  weight  of  the  mass  of  water  in  motion. 

W  v  2 
Then  -       -  is  the  energy  of  the  mass,  and  this  energy  is 

expended  in  opening  the  valve  at  B,  forcing  the  water  into  the 
air-chamber,  compressing  the  air,  and  finally  causing  the  eleva- 
tion of  a  weight  W^  of  the  water  through  a  vertical  distance  k '. 

Let  hf  be  the  head  consumed  in  frictional  and  other  hy- 
draulic resistances. 

Then 

W,(h'  +  hf}  =  the  actual  work  done  =  — '  -. 

This  equation  shows  that,  however  great  h'  may  be,  W^  has 
a  definite  and  positive  value,  and  therefore  water  may  be  raised 
to  any  required  height  by  the  hydraulic  ram. 

WJt' 

The  efficiency  of  the  machine  =      2    ,  and  may  be  as  much 

11 

as  66  per  cent  if  the  machine  is  well  made. 

3.  Pressure-engines.— The  energy  required  to  drive  a  press- 
ure-engine is  usually  supplied  by  means  of  steam-pumps,  but 
an  accumulator  is  often  interposed  between  the  pumps  and  the 
motor  in  order  to  store  up  the  pressure  energy  of  the  water. 
Indeed,  it  is  perhaps  to  the  introduction  of  the  accumulator 
that  the  success  of  hydraulic  transmission  is  especially  due. 
Its  cost,  however,  only  allows  of  its  use  in  cases  where  the 
demand  for  energy  is  for  short  intervals  of  time. 

In  its  simplest  form  the  accumulator  is  merely  a  vertical 
cylinder  into  which  the  water  is  pumped  and  from  which  it  is 
then  discharged  by  the  descent  of  a  heavily  loaded  piston. 
The  water-pressure  thus  developed  in  ordinary  hydraulic  ma- 
chinery is  from  700  to  800  Ibs.  per  square  inch,  but  in  riveting 
and  other  similar  machinery  pressures  of  1500  Ibs.  per  square 
inch  and  upwards  are  often  employed. 

Fig.  134  represents  an  accumulator  designed  by  Tweddell 
for  these  higher  pressures. 


216 


HYDRAULICS. 


The  loaded  cylinder  A  slides  upon  a  fixed  spindle  B. 
The  water  enters  near  the  base,  passes  up  the  hollow  spindle, 
and  fills  the  annular  space  surrounding  the  spindle.  Thus 


FIG.  134- 

the  whole  of  the  weight  is  lifted  by  the  pressure  of  the  water 
upon  a  shoulder  C.  The  water  section  being  small,  any  large 
demand  for  water  will  cause  the  loaded  cylinder  to  fall  rapidly, 
so  that  when  it  is  brought  to  rest  there  will  be  a  considerable 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    2 1/ 

increase  of  pressure  which  is  of  advantage  in  punching,  rivet- 
ing, etc. 

Let  Wbe  the  weight  of  the  loaded  cylinder. 

Let  /'"be  the  friction  of  each  of  the  two  cup-leathers. 

Let  T-J  be  the  radius  of  the  cylinder,  rt  the  radius  of  the 
spindle. 

Let  h  be  the  height  of  the  column  of  water  above  the  pipe  D. 

Let  w  be  the  specific  weight  of  the  water. 

Then/j,  the  intensity  of  the  pressure  in  D  when  the  cylinder 
is  rising, 

W+2F 

=  Wk  -f-      (     a  __ 5rt 

and  /, ,  the  intensity  of  the  pressure  in  D  when  the  cylinder  is 
falling, 

W-2F 


Hence  an  approximate   measure  of   the  variation    of   the 
pressure  is  pl  —p^  —  —, — ^ r. ,  which  ordinarily  varies  from 

about  ifo  of  the  pressure  for  a  i6-in.  ram  to  4$  for  a  4-in.  ram. 

In  a  direct-acting   pressure-engine  let  A  be  the  sectional 
area  of  the  working  cylinder  (Fig,  135). 

Let  a  be  the  sectional  area  of  the  supply- 
pipe. 

Let  A  =  na. 


Let  IV  be  the  weight  of  the  water,  piston,  FJG-  '35- 

and  other  reciprocating  parts  in  the  working  cylii.der. 

Let  /  be  the  length  of  the  supply  pipe. 

Let  f  be  the  acceleration  of  the  piston.     Then  nf  is  the 
acceleration  of  the  water  in  the  supply-pipe. 

The  force  required  to  accelerate  the  piston 


218  HYDRAULICS. 

and  the  corresponding  pressure  in  feet  of  water 

W  f 
~~wAg' 

The  force  required  to  accelerate  the  water  in  the  supply 
pipe 

wal 

•      :  =  ^nf' 

and  the  corresponding  pressure  in  feet  of  water 


A. 
Similarly,  if  /'  is  the  length  of  the  discharge-pipe  and  — 

its  sectional  area,  the  pressure-head  due  to  the  inertia  of  the 
discharge-water 


Hence  the  total  pressure  in  feet  of  water  required  to  over- 
come inertia  in  the  supply-pipe  and  cylinder 


W 

The  quantity  -— ;-)-#/ has  been  designated  the  length  of 

working  cylinder  equivalent  to  the  inertia  of  the  moving  parts. 
Let  the  engine  drive  a  crank  of  radius  r,  and  assume  that  the 
velocity  V  of  the  crank-pin  is  approximately  constant.  Then 
the  acceleration  of  the  piston  when  it  is  at  a  distance  x  from 
its  central  position 

F2 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    21 9 

and  the  pressure  due  to  inertia 


wA^ 

Let  v  be  the  velocity  of  the  piston  in  the  working  cylinder. 

Let  u  be  the  velocity  of  the  water  in  the  supply-pipe. 

Let  h  be   the  vertical  distance  between  the  accumulator- 
ram  and  the  motor. 

Let/0  be  the  unit  pressure  at  the  accumulator-ram. 

Let/  be  the  unit  pressure  in  the  working  cylinder. 

Then 

/0       &a         _  /       V*        (  losses  due  to  friction, sudden  changes 
w       2g         ~~  w      2g       \      of  section,  etc. 

Thus 


A — t  —  v- -11  +  losses. 

W  2g 


V    —  U 

The  term 1-  losses  may  be  approximately  expressed 

o 

v1 
in  the  form  K—  ,  AT  being  the  coefficient  of  hydraulic  resistance. 

Hence 


w  2g 


the  term  h  being  disregarded  as  it  is  usually  very  small  as 

compared  with  —  . 

w 

Thus  the  total  pressure-head  in  feet  required  to  overcome 
inertia  and  the  hydraulic  resistances 


and  is  represented  by  the  ordinate  between  the  parabola  ced 


220 


HYDRA  ULICS. 


and  the  line  ab  in  Fig.  136,  in  which  afgb  is  a  rectangle,  ab 
representing  the  stroke  2r, 


ac  =  oa  — 


the  pressure  due  to  inertia  at  the  end  of  the  stroke,  and 

F2 


the  pressure  required  to  overcome  the  hydraulic  resistances  at 
the  centre  of  the  stroke. 


9 


FIG.  136. 

The  ordinate  between  the   parabola  fmg  and  the  line  fg 
represents  the  back  pressure,  which  is  necessarily  proportional 

Fa 
to  the  square  of  the  piston-velocity,  i.e.,  to  —(r*  —  x*}.     Hence 

the  effective  pressure-head  on  the  piston,  transmitted  to  the 
crank-pin,  is  represented  by  the  ordinate  between  the  curves 
amg  and  ced.  The  diagram  shows  that  the  pressure  at  the 
end  of  the  stroke  is  very  large  and  may  become  excessive.  It 
is  therefore  usual  to  introduce  relief-valves  or  air-vessels  to 
prevent  violent  shocks.  In  certain  cases,  however,  as,  e.g.,  in 
a  riveting-machine,  a  heavy  pressure  at  the  end  of  the  stroke, 
just  where  it  is  most  needed  to  close  the  rivet,  is  of  great 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   221 

advantage,  and  therefore  the  inertia  effect  is  increased  by  the 
use  of  a  supply-pipe  of  small  diameter  and  an  accumulator 
with  a  small  water  section  (Fig.  134). 

The  effective  pressure  should  be  as  great  as  possible,  and 
therefore  the  pressures  due  to  inertia  and  frictional  resistance, 
and  the  back  pressure,  which  are  each  proportional  to  v*,  should 
be  as  small  as  possible,  and  hence  it  is  of  importance  to  fix  a 
low  value  for  the  speed  of  the  piston,  which  in  practice  rarely 
exceeds  80  ft.  per  minute.  The  exhaust  port  should  also  be 
made  of  large  area,  as  the  back  pressure  diminishes  as  the  area 
of  the  port  increases. 

By  equation  I, 


(3) 


This  speed  v  can  be  regulated  at  will  by  the  turning  of  a 
cock,  as  in  this  manner  the  hydraulic  resistances  may  be  in- 
definitely increased. 

Let  the  engine  be  working  steadily  under  a  pressure  Pt  and 
let  v0  be  the  speed  of  steady  motion.  Then 


and 


_  j       useful  resistance  overcome  by  the  piston 

(  +  friction  between  piston  and  accumulator-cylinder. 

If  P  is  diminished,  the  speed  VQ  will  be  slightly  increased, 
but  in  no  case  can  it  exceed, 


4.  Losses  of  Energy. — The  losses  may  be  enumerated  as 
follows : 

(a)  The  Loss  L^  due  to  Piston-friction. — It  may  be  assumed 
that  piston-friction  consumes  from  10  to  20  per  cent  of  the 
total  available  work. 


222  HYDRA  ULICS. 

(b)  The  Loss  Z,  due  to  Pipe-friction.  —  The  loss  of  head  in 
the  supply-pipe  of  diameter  </, 


The  loss  of  head  in  the  discharge-pipe  of  diameter  d^ 


Hence  the  total  loss  of  head  in  pipe-friction  is 

Ml      (nJ 
L'-4f--  — 


The  loss  in  the  relatively  short  working  cylinder  is  very 
small  and  may  be  disregarded. 

(c)  The  Loss  La  due  to  Inertia.  —  The  work  expended  in 
moving  the  water  in  the  supply-pipe 

wA     v* 
gn      ~2~' 

and  in  moving  the  water  in  the  discharge-pipe 

_  wA   ,,  i?_ 
~       1       ~ 


The  total  work  thus  expended 

/,//      l'\v* 
=  wA(--\-  —  }  —  , 
\n  '   ril  2g> 


and  it  may  be  assumed  that  nearly  the  whole  of  this  is  wasted. 
Hence  the  corresponding  loss  of  head  is 


~" 


_/          /'W      _W_ll_         ^_\^__       X 

n    '   ri)~2     ~~  ^2r\n  "•  ~n'}  ^~~     ~2% 


A2r  \n    '   ri2g  ~~  2rn   •    n'     g~~      2g 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    22$ 

(d)  The  Loss  L4  due  to  Curves  and  Elbows. — The  losses  due 
curves  and  elbows  may  be  expressed  in  the  form 

A  =/4—  (Chap.  Ill,  Art.  6). 

(e)  The  loss  L6  due  to  sudden  Changes  of  Section. — The  loss 
of  head  in  the  passage  of  the  water  through  the   ports  may 

be  expressed  in  the  form/' . 

The  loss  occasioned  by  valves  may  also  be  expressed  by 

/f/ 
. 

Thus  the  total  loss  is 


The  coefficient/"  may  be  given  any  desired  value  between 
O  and  oo  by  turning  a  valve,  so  that  any  excess  of  pressure 
may  be  destroyed  and  the  speed  regulated  at  will. 

(/)  The  Loss  Lt  due  to  the  Velocity  with  which  the  Water 
leaves  the  Discharge-pipe. 


A  = 

Hence 


the  effective  head  ==£-•-  (L^  +  A  -  A  +  A  +  L6  +  £„), 

and  the  efficiency  =  I  -    —  (L,  +  A  +  A  +  L<  +  L>). 

The  volume  of  water  used  per  stroke  is  a  constant  quan- 
tity, and  the  efficiency,  which  may  be  as  great  as  eighty  per 
cent  when  the  engine  is  working  under  a  full  load,  may  fall 
below  forty  per  cent  when  the  load  is  light. 

5.  Brakes. —  Hydraulic  resistances  absorb  energy  which  is 
proportional  to  the  square  of  the  speed.  This  property  has 


224  H  YDRA  ULICS. 

been  taken  advantage  of  in  the  design  of  hydraulic  brakes 
for  arresting  the  motion  of  a  rapidly  moving  mass,  as  a  gun 
or  a  train,  of  weight  W.  In  Fig.  137  the  fluid  is  allowed 
to  pass  from  one  side  of  the  piston  to  the  other  through 
orifices  in  the  piston. 

Let  m  be  the  ratio  of  the  area  of  the  piston  to  the  effective 
area  of  the  orifices. 

Let  v  be  the  velocity  of  the -piston  when  moving  under  a 
force  P. 

Let  A  be  the  sectional  area  of  the  cylinder. 


FIG.  137. 
Then 
the  work  done  per  second  =  Pv 

=  the  kinetic  energy  produced 


and  therefore 


P=  wA(m—  i)2  —  , 


and  is  the  force  required  to  overcome  the  hydraulic  resistance 
at  the  speed  v. 

Let  V  be  the  initial  value  of  v,  and  P,  the  maximum  value 
of  P.     Then 

Pl  =  wA(m  —  i)2— 
*g 

Let  F  be  the  friction  of  the  slide.     Then 


— 

o 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    22$ 

and  Pl  -}-  F  is  the  maximum  retarding  force.  It  would  cer- 
tainly be  an  advantage  if  the  retarding  force  could  be  constant. 
In  order  that  this  might  be  the  case  (m  —  i)v  must  be  con- 
stant, and  therefore  as  v  diminishes  m  should  increase  and  con- 
sequently the  orifice  area  diminish.  Various  devices  have  been 
adopted  to  produce  this  result. 

Assuming  the  retarding  force  to  be  constant,  let  x  be  the 
piston's  distance  from  the  end  of  the  stroke  when  its  velocity 
is  v.  Then 


and  therefore  ^2  is  proportional  to  x. 
But  (m  —  i)v  is  constant. 
Therefore  (m  —  i)  is  inversely  proportional  to 

6.  Water-wheels.  —  Water-wheels  are  large  vertical  wheels 
which  are  made  to  turn  on  a  horizontal  axis  by  water  falling 
from  a  higher  to  a  lower  level.     These  wheels  may  be  divided 
into  three  classes  : 

(a)  Undershot  Wheels,  in  which  the  water  is  received  near 
the  bottom  and  acts  by  impulse. 

(b)  Breast   Wheels,  in  which  the  water  is  received  a  little 
below  the  axis  of  rotation  and  acts  partly  by  impulse  and  partly 
by  its  weight. 

(c)  Overshot  Wheels,  in  which  the  water  is  delivered  nearly 
at  the  top  and  acts  chiefly  by  its  weight. 

7.  Undershot  Wheels.  —  Wheels  of  this  class,  with  plane 
floats  or  buckets,  are  simple  in  construction,  are  easily  kept  in 
repair,  and  were  in  much  greater  use  formerly  than  they  are 
now.     They  are  still  found  in  remote  districts  where  there  is 
an  abundance  of  water-power,  and  are  also  employed  to  work 
floating  mills,  for  which  purpose  they  are  suspended  in  an  open 
current  by  means  of  piles  or  suitably  moored  barges.     They 
are  made  from  10  to  25  feet  in  diameter,  and  the  floats,  which 
are   from   24  to  28  in.  deep,  are  fixed   either  normally  to  the 
periphery  of   the   wheel,  or  with    a   slight  slope  towards  the 
supply-sluice,  the  angle  between  the  float   and    radius   being 


226  HYDRA  ULICS. 

from  1 5°  to  30°.     Generally  from  one  half  to  one  third  of  the 
total  depth  of  float  is  acted  upon  by  the  water. 

Let  Fig.  138  represent  a  wheel  with  plane  floats  working  in 
an  open  current. 


FIG.  138. 

Let  vl  be  the  velocity  of  the  current. 
Let  u  be  the  velocity  of  the  wheel's  periphery. 
Let  Q  be  the  delivery  of  water  in  cubic  feet  per  second. 
The  water  impinges  upon  a  float,  is  reduced  to  relative  rest, 
and  is  carried  along  with  the  velocity  u.     Thus 


the  impulse  =  (#,  —  u), 

o 


and 


wQ 
the  useful  work  per  second  =  -  u(vl  —  u). 

o 

Hence 

wQ    . 
—u(y.  —  u)  ,  x 

^        /*=    •  £  2u(v.  —  U) 

the  efficiency  =  —  —  ^-—^  -  =      v  *  a  -  '-, 


which  is  a  maximum  and  equal  to  —  when  u  =  —  v.. 

^  l 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS.    22/ 

Theoretically,  therefore,  the  wheel  works  to  the  best  advan- 
tage when  the  velocity  of  its  periphery  is  one  half  of  the  cur- 
rent velocity.  Even  then  its  maximum  theoretic  effect  is  only 
50$,  and  in  practice  this  is  greatly  reduced  by  frictional  and 
other  losses,  so  that  the  useful  effect  rarely  exceeds  30$. 
Undershot  wheels  with  plane  floats  are  cumbrous,  have  little 
efficiency,  and  should  not  be  used  for  falls  of  more  than  5  feet. 

Again,  let  A  be  the  water-area  of  a  float,  and  w  be  the 
specific  weight  of  the  water. 

wQ  is  somewhat  less  than  wAv^  ,  as  there  will  be  an  escape 
of  water  on  both  sides  of  the  float. 

Let  wQ  =  kwAvlt  k  being  some  coefficient  (<  i)  to  be 
oletermined'by  experiment.  Then 


^ 
the  useful  work  per  second  =  kAw—  l—  (yl  —  u), 

o 

kA 
and  its  maximum  value  =  -  v.w. 


According  to  Bossut's  and  Poncelet's  experiments  a  mean 

A  *y 

value  of  k  is  —  ,  and  the  best  effect  is  obtained  when  u  =  -vl  , 

the  corresponding  useful  work  being  —  —  -  -  -  and  the  effi- 

48 

ciency  —  , 
125 

8.  Wheels  in  Straight  Race.—  Generally  the  water  is  let 
on  to  the  wheel  through  a  channel  made  for  the  purpose,  and 
closely  fitting  the  wheel,  so  as  to  prevent  the  water  escaping 
without  doing  work.  For  this  reason  also,  the  space  between 
the  ends  of  the  floats  in  their  lowest  positions  and  the  channel 
is  made  as  small  as  is  practicable  and  should  not  exceed  2  in. 
Hence  /&,  and  therefore  also  the  efficiency,  will  be  increased. 
Assume  the  channel  to  be  of  a  uniform  rectangular  section  and 
to  have  a  bed  of  so  slight  a  slope  that  it  may  be  regarded  as 
horizontal  without  sensible  error. 


228 


HYDRA  ULICS. 


The  wheel  is  usually  from  24  to  48  ft.  in  diameter,  with  24 
to  48  floats,  either  radial  or  inclined.  The  floats  are  12  to  20 
inches  deep,  or  about  2\  to  3  times  the  depth  of  the  approach- 
ing stream.  The  fall  should  not  exceed  4  ft.  Let  the  floats 
be  radial,  Fig.  139. 


FIG.  139. 

Let  hl  be  the  depth  of  the  water  on  the  up-stream  side  of 

the  wheel. 
Let  //,  be  the  depth  of  the  water  on  the  down-stream  side 

of  the  wheel. 

Let  £,  be  the  width  of  the  race. 
The  impulse  =  impulse  due  to  change  of  velocity 

-|-  impulse  due  to  change  of  pressure 


g  2 

and  the  useful  work  per  second 


=  impulse  X  u  =  ^u(v,  -  u)  +  ^  -  *•)«, 

g  2    Vtf,         -Ul 

The  second  term  is  negative,  since  h^  >  /i,  ,  and  tne  maxi- 
mum theoretic  efficiency  may  be  easily  shown  to  be  <.5. 
Three  losses  have  been  disregarded,  viz.  : 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    229 

(i)  The  loss  of  Ql  cubic  feet  of  the  deeper  fluid  elements 
which  do  not  impinge  upon  some  of  the  foremost  floats. 
According  to  Gerstner, 

o  --=c-2( 


cQ(  *'  V 

*,'U  -  u) ' 


.72,  being  the  number  of  the  floats  immersed,  and  c  being  -J  or 
v  according  as  the  bottom  of  the  race  is  straight  or  falls 
.abruptly  at  the  lowest  point  of  the  wheel. 

(2)  The  loss  of  <22  cubic  feet  of  water  which  escape  between 
the  wheel  and  the  race-bottom. 

Approximately,  the  play  at  the  bottom  may  be  said  to  vary 
from  a  minimum,  sl  =  BC,  when  a  float  AB  is  in  its  lowest 
position,  Fig.  140,  to  a  maximum,  BlCl  =  CD=£^Ct,  when 


FIG.  140. 


two  floats  AlBl  ,  A^Bs  are  equidistant  from  the  lowest  position, 
Fig.  140.     Thus  the  mean  clearance 


=  J(25,  +  BD)  =  5,  +-,  nearly, 
•rl  being  the  wheel's  radius. 


230  HYDRA  ULICS. 

But  -  -  =  distance  between  two  consecutive  floats 

ft 

=  2  .  B^D,  very  nearly, 
n  being  the  total  number  of  floats.     Hence 


a 

and  therefore  the  mean  clearance  =  Sl  -\  ---  —  *. 

Again,  the  difference  of  head  on  the  up-stream  and  down 
stream  sides 


and  the  velocity  of   discharge,  vd,  through   the   clearance   is 
given  by  the  equation 


Hence 


Introducing  .7  as  a  coefficient  of  hydraulic  resistance, 
^  .  /          I  TrVA 

a  =.7,+--^ 


If  the  depth  of  the  stream  is  the  same  on  both  sides  of  the 
wheel,  i.e.,  if  h,  =  &t,  then 


(3)  The  loss  of  03  cubic  feet  of  water  which  escape  between 
the  wheel  and  the  race-sides. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS. 

Let  Ja  be  the  clearance  on  each  side.     Then 


.7  being  a  coefficient  of  hydraulic  resistance. 

Finally,  if   f^lbs.  is  the  weight  on  the  wheel-journals,  the 
loss  due  to  journal  friction 


/*  being  the  journal  coefficient  of  friction,  and  p  the  journal 
radius. 

Thus  the  actual  delivery  of  the  wheel  in  foot-pounds 


These  wheels  are  most  defective  in  principle,  as  they  utilize 
only  about  one  third  of  the  total  available  energy.  They  may 
be  made  to  work  to  somewhat  better  advantage  by  introducing 
the  following  modifications: 

(a)  The  supply  may  be  so  regulated  by  means  of  a  sluice- 
board,  that  the  mean  thickness  of  the  impinging  stream  is  about 
6  or  8  inches.  If  the  thickness  is  too  small,  the  relative  loss  of 
water  along  the  channel  will  be  very  great.  If  the  thickness  is 
too  great,  the  floats,  as  they  emerge,  will  have  to  raise  a  heavy 
weight  of  water.  The  sluice-board  is  inclined  at  an  angle  of 
30°  to  40°  to  the  vertical,  so  that  the  sluice-opening  may  be  as 
near  the  wheel  as  possible,  thus  diminishing  the  loss  of  head 
due  to  channel  friction,  and  is  rounded  at  the  bottom  to  pre- 
vent a  contraction  of  the  issuing  fluid.  Neglecting  frictional 
losses,  etc., 

f  i     re  /->/rr  .   v?       u*\        (  loss  of  energy 

the  useful  effect  =  wQ[H-\--^  --    —  J   ,  _     f7 

\          2£"      2gl        (  due  to  shock 


g 


232  HYDRA  ULICS. 

H  being  the  difference  of  level  between  the  point  at  which  the 
water  enters  the  wheel  and  the  surface  of  the  water  in  the  tail- 
race,  i.e.,  the  fall.  H  is  usually  very  small  and  may  be  negative. 
If  the  vanes  are  inclined,  the  resistance  to  emergence  is  not 
so  great,  and  the  frictional  bed  resistance  between  the  sluice 
and  float  is  practically  reduced  to  nil.  With  a  straight  bed  and 
small  slope  (i  in  10)  the  minimum  convenient  diameter  of 
wheel  is  about  14  ft. 

(b)  The  bed  of  the  channel  for  a  distance  at  least  equal  to 
the  interval  between  two  consecutive  vanes  may  be  curved  to  the 
form  of  a  circular  arc  concentric  with  the  wheel,  with  the  view 
of  preventing  the  escape  of  the  water  until  it  has  exerted  its 
full  effect  upon  the  wheel.     When  the  bed  is  curved,  the  mini- 
mum convenient  diameter  of  wheel  is  about  10  ft. 

An  undershot  wheel  with  a  curb  is  in  reality  a  low  breast- 
wheel,  and  its  theory  is  the  same  as  that  described  in  Arts.  13 
and  14. 

(c)  The  down-stream  channel  may  be  deepened  so  that  the 
velocity  of  the  water  as  it  flows  away  becomes  >  vr     The  im- 
pulse due  to  pressure  is  then  positive,  which  increases  the  useful 
work  and  therefore  also  the  efficiency. 

(d)  The  down-stream  channel  may  be  widened  and  a  slight 
counter-inclination    given    to   the  bed.     What  is  known  as  a 
standing-wave  is  then  produced,  in  virtue  of  which  there  is  a 
sudden  rise  of  surface-level  on  the  down-stream  side  above  that 
on  the  up-stream  side.     This  allows  of  the  wheel  being  lowered 
by  an  amount  equal  to  the  difference  of  level  between  the  sur- 
faces of  the  standing-wave  and  of  the  water-layer  as  it  leaves 
the  wheel,  thus  giving  a  corresponding  gain  of  head. 

(e)  The  introduction  of  a  sudden  fall  has  been  advocated 
in  order  to  free  the  wheel  from   back-water,  but  it   must  be 
borne  in  mind  that  all  such  falls  diminish  the  available  head. 

Thus  undershot  wheels  with  plane  floats  have  little  effect 
because  of  loss  of  energy  by  shock  at  entrance  and  the  loss  of 
energy  carried  away  by  the  water  on  leaving  the  floats.  These 
losses  have  been  considerably  modified  in  Poncelet's  wheel, 
which  is  often  the  best  motor  to  adopt  when  the  fall  does 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    233 

not  exceed  6  ft.,  and  which,  in  its  design,  is  governed  by  two 
principles  which  should  govern  every  perfect  water-motor,  viz. : 

(1)  That  the  loss  of  energy  by  shock  at  entrance  should  be 
a  minimum. 

(2)  That  the  velocity  of  the  water  as  it  leaves  the  wheel 
should  be  a  minimum. 

The  vanes  are  curved  and  are  comprised  between  two 
crowns,  at  a  slightly  greater  distance  apart  than  the  vane- 
width  ;  the  inner  ends  of  the  vanes  are  radial,  and  the  water 
acts  in  nearly  the  same  manner  as  in  an  impulse  turbine. 

First.  Assume  that  the  outer  end  of  a  vane  is  tangential 
to  the  wheel's  periphery,  that  the  impinging  layer  is  infinitely 
thin,  and  that  it  strikes  a  float  tangentially. 

Let  #/(Fig.  141)  be  a  float,  and  aq  the  tangent  at  a. 

The  velocity  of  the  water  relatively 
to  the  float  =  vl  —  u. 

The  water,  in  virtue  of  this  velocity? 
ascends    on    the    bucket    to    a    height 

("     -    V" 
pq  — ,    then    falls    back     and  FlG   I4I 

<§ 

leaves  the  float  with  the  relative  velocity  V1  —  u  and  with  an 
absolute  velocity  vl  —  2u.  This  absolute  velocity  is  nil  when 
the  speed  of  the  wheel  is  such  that  u  =  %i\,  and  the  theoreti- 

i  v  3 

cal  height  of  a  float  is/0  = -.     The  total  available  head  is 

42£- 

thus  changed  into  useful  work,  and  the  efficiency  is  unity,  or 
perfect. 

Taking  R  as  the  mean  radius  of  the  crown  and  ul  as  the 
corresponding  linear  velocity,  the  mean  centrifugal  force  on 

•each  unit  of  fluid  mass  is  -~  and  acts  very  nearly  at  the  direc- 
tion of  gravity,  so  that  the  height  pq  of  a  float  may  be 
approximately  expressed  in  the  form 


'R 


234 


HYDRA  ULICS. 


V  being  the  velocity  with  which  the  water  commences  to  rise 
on  the  float. 

Practically,  however,  the  float  is  not  tangential  to  the  pe- 
riphery at  a,  as  the  water  could  not  then  enter  the  wheel.  Also 
the  impinging  water  is  of  sensible  thickness,  strikes  the  periph- 
ery at  some  appreciable  angle,  and  in  rising  and  falling  on  the 
floats  loses  energy  in  shocks,  eddies,  etc. 

Let  the  water  impinge  in  the  direction  ac,  Fig.  142,  and 
take  ac  =  v^ 

Take  ad  in  the  direction  of  and  equal  to  «,  the  velocity  of 
the  wheel's  periphery. 

Complete  the  parallelogram  bd. 

Then  cd  =  ab  =  V  is  the  velocity  of  the  water  relatively  to 
the  float. 

That  there  may  be  no  shock  at  entrance,  ab  must  be  a  tan- 
gent to  the  vane  at  a. 


FIG.  142. 

Again,  the  water  leaves  the  vane  in  the  direction  of  ba  pro- 
duced, and  with  a  relative  velocity  ae  —  ab  =  V. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    23 $ 


Complete    the   parallelogram   de.     Then  ag(=.  v^  is   the 
absolute  velocity  of  the  water  leaving  the  wheel. 
Evidently  cdg  is  a  straight  line. 

Let  the  angle  cad  =  y,  and  the  angle  bad =  n  —  a. 
From  the  triangle  adc, 

V*  =  v*  -f  u*  —  2v ji  cos  Y  I     •     •      •      •      (i) 
v?  =  V*  -j~  u*  —  2  Vu  cos  a  ;      ....     (2) 

V      sin  Y 

v,       sin  OL  **/ 


From  the  triangle  adg, 
By  equations  I,  2,  and  4, 

^,8    ^  rr  rra  / 

— —  =  —  2  Vu  cos  #  =  vl   —  V    —  u  =  2u(vl  cos  Y  —  u\ 

2 

Therefore  the  useful  work  per  second 

=  ^2U  fa cos  y  - u) (s> 

wQ  v?  cos8  Y 
This    is   a    maximum    and    equal    to when 

V.  COS  Y  rr     • 

u -,  and  the  maximum  emciency  is  cos  y,     Hence^ 

too,  by  equations  I  and  3, 

tan  (n  —  a)  =  2  tan  y (6) 


Also, 

VR  sin 


,  by  equation  6. 


u       sin  (a  -\-  y}       cos  (n  —  a] 

The  efficiency  is  perfect  if  y  is  nil,  and  therefore  a  =  1 80°. 
Practically  this  is  an  impossible  value,  but  the  preceding  cal- 
culations indicate  that  ;  should  not  be  too  large  (usually 
<  30°),  and  that  the  speed  of  the  wheel  should  be  a  little  less 
than  one  half  of  the  velocity  of  the  inflowing  stream. 


236 


HYDRA  ULICS. 


Take  y  =  15°  as  a  mean  value.     Then 

u  =  vt  X  .484,  and  the  efficiency  =  .993. 

Actually  the  efficiency  does  not  exceed  68  per  cent.  In- 
deed it  must  be  borne  in  mind  that  the  theory  applies  to  one 
elementary  layer  only,  say  the  mean  layer,  and  that  all  the 
other  layers  enter  the  wheel  at  angles  differing  from  15°,  thus 
giving  rise  to  "  losses  of  energy  in  shock."  The  losses  of 
energy  in  frictional  resistance,  eddy  motion,  etc.,  in  the  vane 
passages,  have  also  been  disregarded.  The  layers  of  water, 
flowing  to  the  wheel  under  an  adjustable  sluice  and  with  a 
velocity  very  nearly  equal  to  that  due  to  the  total  head,  may 
be  all  made  to  enter  at  angles  approximately  equal  to  15°,  and 
the  corresponding  losses  in  shock  reduced  to  a  minimum  by 
forming  the  course  as  follows : 

The  first  part  of  the  course  FG,  Fig.  143,  is  curved  in  such 
a  manner  that  the  normal pqr  at  any  point/  makes  an  angle 
of  15°  with  the  radius^.  The  water  moves  sensibly  parallel 
to  the  bottom  FG,  and  therefore  in  a  direction  at  right  angles 


FIG.  143. 

to/r.  Hence  at  q  the  direction  of  motion  makes  an  angle  of 
15°  with  the  tangent  to  the  wheel's  periphery.  If  or  is  drawn 
perpendicular  to/r,  then  or  =  oq  sin  15°  =  a  constant. 

Thus  the  normal  pqr  touches  at  r  a  circle  concentric  with 
the  wheel  and  of  a  certain  constant  diameter. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    2$? 

The  initial  point  F  of  the  curve  FG  is  the  point  in  which 
the  tangent  to  this  circle,  passing  through  the  upper  edge  of 
the  sluice-opening,  cuts  the  bed  of  the  supply-channel. 

If  t  is  the  thickness  (or  depth  of  sluice-opening)  and  b  the 
breadth  of  the  layer  of  water  as  it  leaves  the  sluice,  then 

Q  =  btv,  , 
and  according  to  Grashof 


H  being  the  available  fall. 

The  thickness  should  not  exceed  12  to  15  inches,  and  is 
generally  from  8  to  10  inches. 

Neglecting  float  thickness,  the  capacity  of  the  portion  of 
the  wheel  passing  in  front  of  the  entering  stream  per  second 
=  bdu^  ,  very  nearly. 

Only  a  portion  of  this  space  can  be  occupied  by  the  water, 
so  that 

Q  —  mbdul  , 

m  being  a  fraction  whose  value  may  be  taken  to  be  J  or  f  „ 
Hence 

mbdul  —  btv^  , 
and  therefore 

u.      md          u. 
t  =  md—  =  —  cos  y  — 

Vl  2  r    U 

md          R 
=  —  cos  v—  . 
2         rr, 

According  to  Morin, 

r,  =  2d  to  $d. 


The  mean  velocity  at  entrance  =  cv<    2g(H  —  £/),  an  aver- 
age  value  of  cv  being  .9. 

Thus  \it  =       , 


HYDRAULICS. 


The  diameter  of  the  wheel  is  often  taken  to  be 

The  area  of  the  sluice-opening  is  usually  from  \\bt  to  i.^bt. 

The  inside  width  of  the  wheel  is  about  (b  +  J)  ft. 

The  water  should  not  rise  over  the  top  of  the  buckets,  and 
in  order  to  prevent  this  the  depth  of  the  shrouding  is  from  J// 
to  \H. 

If  A  is  the  angle  subtended  at  the  centre  O  of  the  wheel  by 
the  water-arc  between  the  point  of  entrance  A  and  the  lowest 


point  €,  Fig.  144,  of  the  wheel,  and  if  Aq'  is  drawn  horizontally, 
then  Aq'  is  approximately  the  height  of  the  float,  and  the 
theoretic  depth  d  of  the  crown  is  given  by 

'  +  OC  -  Oq' 


=  AC  =  Aqf  +Cq'  = 


In  practice  it  is  usual  to  increase  this  depth  by  /,  the  thick- 
ness of  the  impinging  water-layer. 
Again, 

2       V"1 
d  — s  -f  r,(i  —  cos  A)  -f-  a  few  inches,  approximately. 


The  buckets  are  usually  placed  about  I  ft.  apart,  measured 
along  the  circumference,  but  the  number  of  the  buckets  is  not 
a  matter  of  great  importance.  There  are  generally  36  buckets 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    239 

in  wheels  of  10  to  14  ft.  diameter,  and  48  buckets  in  wheels  of 
20  to  23  ft.  diameter. 

It  may  be  assumed  that  the  water-arc  is  equally  divided  by 
the  lowest  point  C  of  the  wheel,  so  that 

the  length  of  the  water-arc  =  2\r  =  2uT, 

T  being  the  time  of  the  ascent  or  descent  of  the  water  in  the 
bucket. 

In  the  middle  position,  the  upper  end  of  the  bucket  should 
be  vertical,  and  if  the  float  is  in  the  form  of  a  circular  arc,  its 
radius  r'  =  d  sec  (it  —  a\  a  being  the  angle  between  the 
bucket's  lip  and  the  wheel's  periphery. 

The  time  of  ascent  or  descent  is  also  given  by 


where  sin  fy  =  I/cos  (it  —  a). 

9.  Efficiency  corresponding  to  a  Minimum  Velocity  of 
Discharge  (V2).  —  From  Fig.  142, 


ao  (=  \ag)  _    sin  y   __  £Qa) 
ad  sin  aod         u 

Hence  for  any  given  values  of  u  and  y,  vz  is  a  minimum 
when  sin  aod  is  greatest,  that  is,  when  aod  =  90°,  or  ag  is  at 
right  angles  to  de.  Then  also  ad  =  ae  =  ab,  or  u  =  V,  and  ac 
bisects  the  angle  bad.  Thus, 

i71  =  2u  cos  y     and     v^  —  2u  sin  y. 
The  useful  work 

W  v?  —  v?       W  WV/cos  2y 

=  —  .  -'  -  '-  =  —2u*  cos  2y  =  --  5-  -  £, 

g  2  g  g    2    COS'  Y 

The  total  available  work 


240  H  YDRA  ULICS. 

Therefore  the  efficiency 

cos  2v 

- 


Ex.  —  If  y  =  15°,  the  efficiency  =  .928  and  u  =  . 
In  practice  the  best  value  of  u  is  found  to  lie  between. 
and  .60^. 

The  horse-power  of  the  wheel 


rf  being  the  efficiency  with  an  average  value  of  60$. 

Although,  under  normal  conditions  of  working,  the  effi- 
ciency of  a  Poncelet  wheel  is  a  little  less  than  that  of  the  best 
turbines,  the  advantage  is  with  the  former  when  working  with 
a  reduced  supply. 

10.  Form  of  Bucket  —  The  form  of  the  bucket  is  arbitrary, 
and  may  be  assumed  to  be  a  circular  arc.  In  practice  there 
are  various  methods  of  tracing  its  form. 

METHOD  I  (Fig.  145),     The  tangent  am  to  the  bucket  at  a 


FIG.  145. 

makes  a  given  angle  a  with  the  tangent  at  a  to  the  wheel's 
outer  periphery.     The  radius  of\s  also  a  tangent  to  the  bucket 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    24! 

at/!  If  the  angle  aof\s  known  the  position  of  f  on  the  inner 
periphery  is  at  once  fixed,  and  the  form  of  the  bucket  can  be 
easily  traced. 

Let  the  angle  aof—x.     Join  af  and  let  the  tangents  to  the 
bucket  at  a  and /meet  in  m.     Then 
the  angle  oam  =  a  —  90°. 

"      oma  —  1 80°  —  oam  —  aom  =  270°  —  a  —  x. 
"      mfa  =  the  angle  maf  —  £(180°  —  fmd) 

=  "+-*- 45-  ' 

Let  rlt  r^  be  the  radii  of  the  outer  and  inner  peripheries  of 
the  wheel.     Then 

sin  (f!L±£  _  45°) 
rl oa  sin  of  a  sin  mfa  \      2  / 


of       sin  oaf       sin  oaf 


sin  (^£-45*) 


since  the  angle  oaf '  =•  oam  —  maf  '=  -       45°. 

Hence 


r.  — 


X 

tan  - 
2 


tan    - 


an  equation  giving  ;tr. 

The  point  o'  in  which  the  perpendicular  o'f  to  0/"  meets 
the  perpendicular  o'a  to  am  is  the  centre  of  the  circular  arc 
required  and  o'f(^o'd)  is  the  radius. 

METHOD  II  (Fig.  146).  Take  mad =  150°,  and  in  ma  pro- 
duced take  ak  =  of.  With  k  as  centre  and  a  radius  equal  to 


242 


HYDRAULICS. 


ao  describe  the  arc  of  a  circle  intersecting  the  inner  periphery 
in  the  point  f.  Join  kf,  of,  and  af.  The  two  triangles  aof 
and  akf  are  evidently  equal  in  every  respect,  and  therefore 
the  angle  kaf  is  equal  to  the  angle  of  a.  Drawing  ao'  at  right 
angles  to  ak  and  fo'  tangential  to  the  periphery  at  f,  the  angle 
0'af(=  kaf —  90°)  is  equal  to  the  angle  o'f  a  (=  of  a  —  90°),  and 
therefore  o'a  =  o'f.  Thus  o'  is  the  centre  of  the  circular  arc 
required  and  o'a  (=  o'f)  is  the  radius. 


FIG.  146. 

9- 

METHOD  III  (Fig.  147).  Let  the  bed  with  a  slope  of,  say, 
i  in  10  extend  to  the  point  C,  and  then  be  made  concentric 
with  the  wheel  for  a  distance  CC  subtending  an  angle  of  30° 
at  the  centre  of  the  wheel.  Let  the  mean  layer,  half  way 
between  the  sloping  bed  and  the  surface  of  the  advancing 
water,  strike  the  outer  periphery  at  the  point  /.  Draw  fk 
making  an  angle  of  23°  with  of,  and  take  fk  equal  to  one  half 
or  seven  tenths  of  the  available  fall,  k  is  the  centre  of  the 
circular  arc  required  and  £/is  its  radius. 

II.  Breast-wheels. — These  wheels  are  usually  adopted  for 
falls  of  from  5  to  15  feet,  and  for  a  delivery  of  from  5  to  80 
cubic  feet  per  second. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    243 

The  diameter  should  be  at  least  1 1  ft.  6  in.,  and  rarely  ex- 
ceeds 24  ft.  The  velocity  (u)  of  the  wheel's  periphery  is  gen- 
erally from  3^  ft.  to  5  ft.  per  second,  the  most  useful  average 
velocity  being  about  4^  ft.  per  second. 

The  width  of  the  wheel  should  not  exceed  from  8  to  10  ft. 

It  is  of  great  importance  to  retain  the  water  in  the  wheel 
as  long  as  possible,  and  this  is  effected  by  surrounding  the 


water-arc  with  an  apron,  or  a  curb,  or  a  breast,  which  may  be 
constructed  of  timber,  iron,  or  stone.  Hence,  too,  the  buckets 
may  be  plane  floats,  but  they  should  be  set  at  an  angle  to  the 
periphery  of  the  wheel,  so  as  to  rise  out  of  the  water  with  the 
least  resistance  (Art.  8). 

The  depth  of  a  float  should  not  be  less  than  2.3  ft.,  and  the 
space  between  two  consecutive  floats  should  be  filled  to  at 
least  one  half,  and  even  to  two  thirds,  of  its  capacity.  The 
head  (measured  from  still  water)  over  the  sill  or  lip  should  be 
about  9  in. 

The  play  between  the  outer  edge  of  the  floats  and  the 
curb  varies  from  £  in.  in  the  best  constructed  wheels  to 
2  inches. 

The  distances  between  the  floats  is  from  i^  to  if  times  the 
head  over  the  sill. 


244 


HYDRA  ULICS. 


Breast-wheels  are  among  the  best  of  hydraulic  motors, 
giving  a  practical  efficiency  which  may  be  as  large  as  80 
per  cent. 

12.  Sluices. — The  water  is  rarely  admitted  to  the  wheel 
without  some  sluice  arrangement,  which  may  take  the  form  of 

an  overfall  sluice  (Fig.  148), 
an  underflow  sluice  (Fig.  149), 
or  a  bucket  or  pipe  sluice 
(Fig.  150). 

The  pipe  sluice  is  espe- 
cially adapted  for  a  varying 
supply,  being  provided,  for  a 
certain  vertical  distance,  with 
a  series  of  short  tubes,  so  in- 
clined as  to  ensure  that  the 
water  enters  the  wheel  in  the 
right  direction.  Taking  .85 
as  the  mean  coefficient  of 
hydraulic  resistance  for  these 
tubes,  the  head  kl  required 
to  produce  the  velocity  of 
entrance  z>  is 


and  if  H  is  the  total  available 
fall, 


=  remainder  of  fall  available  for  pressure-work. 

The  profile  AB  in  an  overfall  and  an  underflow  sluice, 
should  coincide  with  the  parabolic  path  of  the  lowest  stream- 
lines of  the  jet.  The  crest  of  the  overfall  should  be  properly 
curved,  and  the  inner  edges  of  the  underflow  opening  should 
be  carefully  rounded  so  as  to  eliminate  losses  due  to  con- 
traction 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS.    24$ 

The  underflow  sluice-opening  should  also  be  normal  to 
the  axis  of  the  jet. 

Let  h^  be  the  head  above  the  crest  of  an  overfall  sluice. 
Then 

2     T.      •' * 

Q  =  -cb, 


b^  being  the  width  of  the  crest  and  c  the  coefficient  of  dis- 
charge.    The  width  bl  is  usually  3  or  4  inches  less  than  the 
width  b  of  the  wheel. 
From  this  equation 


and  the  depth  of  water  over  the  crest  or  lip  is  usually  about 
9  inches. 

Again,  the  head  h^=  CD)  required  to  produce  the  velocity 
vl  at  the  point  of  entrance  B  is 


10 


10  per  cent  being  allowed  for  loss  due  to  friction. 

Thus   the   height  of  the  crest  A   above  B,  the  point  of 
entrance, 


=  AD  =  CD  -  CA  =  h,  - 

ii  *;/  36  V 

10     2g       \2cb^2g)' 


But  BA  is  a  parabola  with  its  vertex  at  A,  and  therefore, 
if  B  is  the  angle  between  the  horizontal  BD  and  the  tangent 
the  parabola  at  B, 


n  •         «      f\  A 

V,  sm  u  1 1  v* 

2g  ~  10     2g 


y 

) 


246  HYDRA  ULICS. 

Also 

v.  sin  26 


The  head  available  for  pressure  work 

=  DE  =  FG  =  H  -  h,. 


Let  a  be  the  angle  between  BT  and  the  tangent  to  the 
wheel's  periphery  at  B.     Then 

a  _f  0  =  the  angle  EOF, 

BO  being  the  radius  to  the  centre  of  the  wheel  and  OFG' 
vertical. 

%  If  the  lowest  point   G'  of  the  wheel  just  clears  the  tail- 
race,  the  head  available  for  pressure  work 

=  H  -  h,  =  FG'  —  OG'  -  OF 

=  rfr  _  cos  BOF)  =  2r,  si 


r,  being  the  radius  to  the  outer  periphery  of  the  wheel. 
If,  again,  the  water  enters  the  wheel  tangentially, 

a  =  o,  and  the  angle  BOF  =  B, 
so  that 

H  -  h,  =  2r,  sin2  -. 

If  the  sluice-opening  is  not  at  the  vertex  of  the  parabola, 
the  axis  of  the  opening  should  be  tangential  to  the  parabola. 

13.  Speed  of  Wheel.  —  The  water  leaves  the  buckets  and 
flows  away  in  the  race  with  a  velocity  not  sensibly  different 
from  the  velocity  u  of  the  wheel's  periphery. 

Let  b  be  the  breadth  of  the  wheel  (Fig.  151). 

Let  x  be  the  depth  of  the  water  in  the  lowest  bucket. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    247 


FIG.  151. 

Allowing  for  the  thickness  of  the  buckets,  the  play  between 
the  wheel  and  curb,  etc., 


Q  =  cbxu, 


c  being  an  empirical  coefficient  whose  average  value  is  about 
.0.     Hence 


10   Q 

u  = jr. 

9    ox 


In  practice  b  is  often  taken  to  be  —  to  — .  It  is  impor- 
tant that  b  should  be  as  small  as  possible  and  hence  x  should 
be  as  large  as  possible,  its  value  being  usually  ij  ft.  to  2  ft. 

It  must  be  borne  in  mind,  however,  that  any  increase  i-n 
the  value  of  x  will  cause  an  increase  in  the  weight  of  water 
lifted  by  the  buckets  as  they  emerge  from  the  race,  and  will 
therefore  tend  to  diminish  the  efficiency. 

14.  Mechanical  Effect.— Theoretically,  the  total  mechan- 
ical effect 


248 


HYDRA  ULTCS. 


H  being  the  fall  from  the  surface  of  still  water  in  the  supply- 
channel  to  the  surface  of  the  water  in  the  tail-race. 
This,  however,  is  reduced  by  the  following  losses: 
(a)  Owing  to  frictional  resistance,  etc.,  there  is  a  loss  of 

v  3 
head  in  the  supply-channel  which  may  be  measured  by  ^-7- 

v  being  approximately  JL  to  TL. 

The  head  required  to  produce  the  velocity  at  entrance,  vl9 


(b)  Let  af,  Fig.   152,  represent  in  direction  and  magnitude 
v,  the  velocity  of  the  water  entering  the  bucket. 


FIG.  152. 

Let  ad,  in  the  direction  of  the  tangent  to  the  wheel's 
periphery,  represent  the  velocity  u  of  the  periphery  in  direction 
and  magnitude. 

Complete  the  parallelogram  bd.  Then  ab  evidently  repre- 
sents the  velocity  V  of  the  water  relatively  to  the  wheel. 
This  velocity  V  is  rapidly  destroyed,  the  corresponding  loss  of 
head  being 

F2        U*-\-V?  —  2UV^  COS  y 


being  the  angle  daf. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    249 

Assuming  that  the  water  enters  the  race  with  the  velocity 
u  of  the  wheel,  the  theoretical  useful  work  per  pound  per 
second  due  to  impact 


u. 
=  -(vl  cos  y  —  u). 

g 

V^ 
If  the  loss  —  is  to  be  a  minimum   for  a  given  speed  of 

o 

wheel, 

v,dv^  —  u  cos  y  .  dvl  =  o,     or     ^  —  u  cos  y.  .     .     (2) 

Hence,  by  equation  I,  V  =  u  sin  7,  and  therefore 

V      df 

tan  y  =  -  =  2 
v,       af 

so  that  for  a  velocity  of  entrance  vt  =  u  cos  y  the  angle  afd 
should  be  90°.  But  this  value  is  inadmissible,  as  the  water 
would  arrive  tangentially  and  consequently  would  not  enter  the 
buckets.  In  Order  that  the  loss  in  shock  at  entrance  may  be  as 
small  as  possible,  ab,  the  direction  of  the  relative  velocity  F, 
should  be  parallel  to  the  arm  xy  of  the  bucket,  and  should 
therefore  be  approximately  normal  to  the  wheel's  periphery. 
This  is  equivalent  to  the  assumption  that  the  water  arrives  in 
a  given  direction  (y)  with  a  given  velocity  (^),  and  that  the 
speed  (?/)  of  the  wheel  is  to  be  such  as  will  make  V  a  mini- 
mum. Thus,  by  equation  I, 

o  —  udu  —  v^  cos  y  .  du,     or     u  =  vl  cos  y, 
and  therefore 

V  =  vl  sin  y. 

Hence  tan  y  =  —  —  -£,  and  therefore  the  angle  adf  =  90°. 
u       ad 


250 


HYDRAULICS. 


In  practice  y  is  generally  30°,  and  the  corresponding  loss  of 


Fa       v?    .  v>    i       if    i 

head  =  —  =  —  sin2  y  =  —  •-.  -  =  —  .  - 


At  point  of  entrance  x  falls  below  y,  the  water  flows  up  the 
inclined  plane  xy,  and  F,  instead  of  being  wholly  destroyed  in 
eddy  motion,  is  partially  destroyed  by  gravity.  This  velocity, 
destroyed  by  gravity,  is  again  restored  to  the  water  on  its 

return,  and  thus  adds  to  the  efficiency 
of  the  wheel. 

It  will  be  found  advantageous  to 
use  curved  or  polygonal  buckets  and 
not  plane  floats.  A  bucket,  for  ex- 
ample, may  consist  of  three  straight 
portions,  ab,  be,  cd,  Fig.  153.  Of  these 
the  inner  portion  cd  shoud  be  radial  ; 
the  outer  portion  ab  is  nearly  normal  to  the  periphery  of  the 
wheel,  and  the  central  portion  be  may  make  angles  of  about 
135°  with  ab  and  cd. 

Disregarding  all  other  losses,  the  theoretical  delivery  of  the 
wheel  in  foot-pounds 


where  h^  =  total  fall  —  fall  (h^  required  to  produce  the  veloc- 
ity v,. 

If  77  be  the    efficiency,  then,  according  to  the  results  of 
Morin's  experiments, 

rf  =  .40  to  .45  if  h^  =  -//"; 
4 

rf  =  .42  to  .49  if  hl  =  —H\ 

rj  =  .47  if  h,  =  -H; 

3 


if  h,  =    ff. 
4 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    2$  I 

(c)  There  is  a  loss  of  head  due  to  frictional  resistance  along 
the  channel  in  which  the  wheel  works. 

Let  /  =  length  of  the  channel  (or  curb). 

Let  t  —  thickness  of  water-layer  leaving  the  wheel. 

Let  b  =  breadth  of  wheel. 

The  mean  velocity  of  flow  in  this  curb  channel  is  approxi- 

mately -u,  and  the  loss  of  head  due  to  channel  friction 


bt       2g 

where/  =  coefficiency  of  friction,  b  -f-  2t  =  wetted  perimeter, 
bt  =  water  area,  and  y  being  30°. 

(d]  There  is  a  loss  of  head  due  to  the  escape  of  water  over 
the  ends  and  sides  of  the  buckets. 

Let  s1  be  the  play  between  the  ends  of  the  buckets  and  the 

channel. 

Let  s^  be  the  play  at  the  sides.     (^,  =  Ja  ,  approximately.) 
Let  zl  ,  #2  ,  .  .  .  zn  be  the  depths  of  water  in  a  bucket  corre- 
sponding to  n  successive  positions   in  its  descent 
from  the  receiving  to  the  lowest  points. 
Let  /a  ,  /a,  ...  ln  be  the  corresponding  water-arcs  measured 

along  the  wheel's  periphery. 

The  orifice  of  discharge  at  end  of  a  bucket  =  bs^ 
The  mean  amount  of  water  escaping  from  a  bucket  over 
its  end 


c  being  the  coefficient  of  discharge. 

The  water  escapes  at  the  sides  as  over  a  series  of  weirs, 
and  the  mean  amount  of  water  escaping  from  a  bucket  over 
the  sides 


252  HYDRAULICS. 

Hence  the  total  loss  of  effect  from  escape  of  water 


per  sec.,  ^  being  the  vertical  distance  between  the  point  of 
entrance  and  the  surface  of  the  water  in  the  tail-race 


__. 

(e)  There  is  a  loss  of  head  due  to  journal  friction. 

Let  W  =  weight  of  wheel. 

Let  wl  =  weight  of  water  on  the  wheel. 

Let   rl  =  radius  of  wheel's  outer  periphery. 

Let    r1  —  radius  of  axle. 

Loss  per  second  of  mechanical  effect  due  to  journal  friction 


r  being  the  coefficient  of  journal  friction. 

There  is  a  loss  of  mechanical  effect  due  to  the  resistance  of 
the  air  to  the  motion  of  the  floats  (buckets),  but  this  is  prac- 
tically very  small,  and  may  be  disregarded  without  sensible 
error.  A  deepening  of  the  tail-race  produces  a  further  loss  of 
effect,  and  should  only  be  adopted  when  back-water  is  feared. 

Hence  the  total  actual  mechanical  effect, 
putting 

Z=bSl(  V^ 


cos 

,s  = 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS.    253 

=wQ    ff-  (i  +  v)        +         fa  cos  r  -  «) 


--*(",  cos  y-u) 


Hence,  for  a  given  value  of  z/,,  the  mechanical  effect  (omit- 
ting the  last  term)  is  a  maximum  when 

«  =  ^  C°S  Y  (=  -433  X  ^  ,  if  r  =  30°). 


In  practice  the  speed  of  the  wheel  is  made  about  one  half 
of  the  velocity  with  which  the  water  enters  the  wheel. 

For  a  given  speed  of  wheel,  and  disregarding  the  loss  of 
effect  due  to  curb  friction,  which  is  always  small,  the  mechani- 
cal effect  is  a  maximum  for  a  value  of  z/,  given  by 

I    ^         t/—w'Z\l  +  v      i   WQ 
—  \wQ  —  c  V2g — 1  — ! — vl  H -u  cos  Y  =  o, 

or 

U  COS  Y 




The  loss  by  escape  of  water,  viz.,  c  V2g—,  varies,  on  an 
average,  from  10  to  15  per  cent  of  the  whole  supply,  so  that 

c  V2g-  varies  from  —  to  2s, 
d  n  10       20 


254  JfYDRA  ULICS. 

15.  Sagebien  Wheels  have  plane  floats  inclined  to  the 
radius  at  from  40°  to  45°  in  the  direction  of  the  wheel's  rota- 
tion. The  floats  are  near  together  and  sink  slowly  into  the 
fluid  mass.  The  level  of  the  water  in  the  float-passages  grad- 


FIG.  154. 

ually  varies  and  the  volume  discharged  in  a  given  time  may 
be  very  greatly  changed.  The  efficiency  of  these  wheels  is 
over  80  per  cent,  and  has  reached  even  90  per  cent.  The 
action  is  almost  the  same  as  if  the  water  were  transferred  from 
upper  to  lower  race,  without  agitation,  frictional  resistance, 
etc.,  flowing  away  without  obstruction,  into  the  tail-race. 

16.  Overshot  Wheels. — These  wheels  are  among  the  best 
of  hydraulic  motors  for  falls  of  8  to  70  ft.  and  for  a  delivery  of 
3  to  25  cub.  ft.  per  second.     They  are  especially  useful  for  falls 
of  12  to  20  ft.     The  efficiency  of  overshot  wheels  of  the  best 
construction  is  from  .70  to  .85. 

If  the  level  of  the  head-water  is  liable  to  a  greater  variation 
than  2  ft.,  it  is  most  advantageous  to  employ  a  pitch-back  or 
high  breast-wheel,  which  receives  the  water  on  the  same  side 
as  the  channel  of  approach. 

17.  Wheel-velocity. — This   evidently   depends   upon    the 
work  to  be  done,  and  upon  the  velocity  with  which  the  water 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   255 

arrives  on  the  wheel.  Overshot  wheels  should  have  a  low 
circumferential  speed,  varying  from  10  ft.  per  sec.  for  large 
wheels  to  3  ft.  per  sec.  for  small  wheels,  and  should  not  be  less 
than  2-J  ft.  per  sec. 

In  order  that  the  water  may  enter  the  buckets  easily,  its 
velocity  should  be  greater  than  the  peripheral  velocity  of  the 
wheel. 

18.  Effect  Of  Centrifugal  Force. — Consider  a  molecule 
of  weight  W  in  the  "  unknown"  surface  of  the  water  in  a 


FIG.  155. 

bucket  (Fig.   155).     At   each  moment  there  is   a  dynamical 
equilibrium    between  the  "  forces"    acting  on   m,  viz.:   (i)  its 


256  HYDRA  ULICS. 

IV 

weight  w\  (2)  the  centrifugal  force  —  coV;  (3)  the  resultant  T 

o 

of  the  neighboring  reactions. 

2V 

Take  MF  =  w,  MG  =  — coV,  and  complete  parallelogram 

o 

FG.     Then  MH  =  T.     The  direction  of  T  is,  of  course,  normal 
to  the  surface  of  the  water  in  the  bucket. 

Let  HM  produced  meet  the  vertical  through  the  axis  O  of 
the  wheel  in  E.     Then 


w_    a 
MG       z**r      FH      OM        r 


MF~        w      ~MF~OE"OE' 
and  therefore 

OB  =*,  = 

GO 

taking  g  =  32  ft.  and  n  being  the  number  of  revolutions  per 
minute. 

Thus  the  position  of  E  is  independent  of  r  and  of  the 
position  of  the  bucket,  so  that  all  the  normals  to  the  water- 
surface  in  a  bucket  meet  in  E,  and  the  surface  is  the  arc  of  a 
circle  having  its  centre  at  E,  or,  rather,  a  cylindrical  surface 
with  axis  through  E  parallel  to  the  axis  of  rotation. 

19.  Weight  of  Water  on  Wheel  and  Arc  of  Discharge.— 
Let  Q  =  volume  supplied  per  sec.,  and  N  =  number  of  buckets. 

Noo 
Then  -    -  =  number  of  buckets  fed  per  sec., 

27T 

and      — —  =  volume  of  water  received  by  each  bucket  per  sec. 
Hence  the  area  occupied  by  the  water  until  spilling  com- 
mences =  , .,   ,  b  being  the  bucket's  width  (=  width  of  wheel 

between  the  shroudings). 

The  water  flows  on  to  the  wheel  through  a  channel  (Fig. 
156),  usually  of  the  same  width  b  as  the  wheel,  and  the 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS. 


supply  is  regulated  by  means  of  an  adjustable  sluice,  which 
may  be  either  vertical,  inclined,  or  horizontal. 

When  the  water  springs  clear  from  the  sluice,  as  in  Fig.  156, 
the  axis  of  the  sluice  should  be  tangential  to  the  axis  of  the 


FIG.  156. 

jet,  and  the  inner  edges  of  the  sluice-opening  should  be  rounded 
so  as  to  eliminate  contraction. 

Let  y,  z  be  the  horizontal  and  vertical  distances  between 
the  sluice  and  the  point  of  entrance. 

Let  T  be  the  time  of  flow  between  the  sluice  and  entrance. 

Let  v0 ,  2\  be  the  velocities  of  flow  on  leaving  the  sluice  and 
on  entering  the  bucket. 

Then 


258  H  YDRA  ULICS. 

and 

V?  =  V*  +  2gZ, 

d  being  angular  deviation  of  point  of  entrance  from  summit, 
and  y  the  angle  between  the  direction  of  motion  of  the  water 
and  the  wheel  at  the  point  of  entrance. 

Assume  the  bed  of  the  channel  to  be  horizontal,  and  the 
sluice  vertical  and  of  the  same,  width  b  as  the  wheel.  The 
sluice  is  also  supposed  to  open  upwards  from  the  bed.  Then 


x  being  the  depth  of  sluice-opening  and  h^  the  effective  head 
over  the  sluice.  This  effective  head  is  about  TVths  of  the  actual 
head. 

Thus,  taking  g=.  32,  •—  =  %xh$  gives  the  delivery  per  foot 

width  of  wheel. 

Taking  .6  ft.  and  3.6  ft.  as  the  extreme  limits  between 
which  hl  should  lie,  and  .2  ft.  and  .33  ft.  as  the  extreme  limits 

between  which  x  should  lie,  then  ~  must  lie  between  the 

o 

limits  1.24  and  5,  and  an  average  value  of  ^  is  3.     Thus  the 

width  of  the  wheel  should  be  on  the  average  ^  — . 

Again,  neglecting  the  thickness  of  the  buckets,  the  capacity 
of  the  portion  of  the  wheel  passing  in  front  of  the  water-sup- 
ply per  second 


=  b<*>  \  —  —  —  -  -  !•  =  Mfafr,  --  J  =  bdrja,  approximately, 


,  ,          Lj 
=  bdu.  —  bd 


30 

r,  being  the  radius  and  ul  the  velocity  of  the  outer  circumfer- 
ence of  the  wheel,  d  the  depth  of  the  shrouding,  and  n  the 
number  of  revolutions  per  minute. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   259 

Only  a  portion,  however,  of  the  space  can  be  occupied  by 
the  water,  so  that  the  capacity  of  a  bucket  is  mubd,  m  being 
a  fraction  less  than  unity  and  usually  -J  or  J.  For  very  high 
wheels  m  may  be  \.  Hence 

,   ,  27tQ 

mbdu.  =  ~=. 

NGO 

Again,  since  the  thickness  of  the  buckets  is  disregarded, 

Nu  — 


Therefore  mdu.  =  ^. 

b 


The  delivery  \^j   per  foot  of  width  must  not  exceed   a 

certain  limit,  otherwise  either  d  or  u  will  be  too  great.  In  the 
former  case  the  water  would  acquire  too  great  a  velocity  on 
entering  the  buckets,  which  would  lead  to  an  excessive  loss  in 
eddy  motion  and  a  corresponding  loss  of  efficiency ;  while  if 
the  speed  u  of  the  wheel  is  too  great  the  efficiency  is  again 
diminished  and  might  fall  even  below  40$. 

The  depth  of  a  bucket  or  of  the  shrouding  varies  from  10 
to  1 6  in.,  being  usually  from  10  to  12  in.,  and  the  buckets  are 
spread  along  the  outer  circumference  at  intervals  of  12  to 
14  inches.  The  number  of  the  buckets  is  approximately  $r  or 
6r,  r  being  the  radius  of  the  wheel  in  feet. 

The  efficiency  of  the  wheel  necessarily  increases  with  the 
number  of  the  buckets,  but  the  number  is  limited  by  certain 
considerations,  viz. :  (a)  the  bucket  thickness  must  not  take  up 
too  much  of  the  wheel's  periphery  ;  (b)  the  number  of  the 
buckets  must  not  be  so  great  as  to  obstruct  the  free  entrance 
of  the  water;  (c)  the  form  of  the  bucket  essentially  affects  the 
number. 

Let  the  bucket,  Fig.  157,  consist  of  two  portions,  an  inner 
portion  be,  which  is  radial,  and  an  outer  portion  cd\  c  being  a 
point  on  what  is  called  the  division  circle.  The  length  be  is 
usually  one  half  or  two  thirds  of  the  depth  d  of  the  shrouding. 


260 


HYDRA  ULICS. 


Take  be  =  \d. 

It  may  also  be  assumed  without  much  error  that  the  water- 
surface  ad  is  approximately  perpendicular  to  the  line  edt  so 
that  the  angle  edc  is  approximately  a  right  angle. 

The  spilling  evidently  commences  when  the  cylindrical  sur- 
face, having  its  axis  at  e  and  cutting  off  from  the  bucket  a 


water-area  equal  to  -~,  passes  through  the  outer  edge  d  of 


Noo 


the  bucket. 


FIG.  157. 

Let  /3  be  the  bucket  angle  cOd. 

Let   0  be  the  inclination  of  Od  to  the  horizon. 

Let  0  be  the  inclination  of  ad  to  the  horizon. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  26 1 

Let  rl  be  the  radius  of  the  outer  periphery. 
Let  R  be  the  radius  of  the  division  circle. 
Let  ra  be  the  radius  of  the  inner  periphery. 
Then 


od^   __  rl  _  sin  0  _         sin  0 

oe  ""^"sin  j  90°—  0+0}  ~  cos  (0+0)' 


and  therefore 


Again, 


Therefore 


sin  0 


af  =  fd  tan  (0  -|-  0),  approximately. 


the  area  dfa=<—  tan  (0  +  0)  =  —  tan  (8  +  0), 

2  2 


where  d  =  rl  —  r2.     Hence 


the  area  abed  =  area  cod  —  area  bof  —  area  ^/iz 


Equations  (i)  and  (2)  give  0  and  0,  and  therefore  the  posi- 
tion of  the  bucket  when  spilling  commences. 

The  bucket  will  be  completely  emptied  when  it  has  reached 
a  position  in  which  cd  is  perpendicular  to  a  line  from  e  to 
middle  point  of  cd,  or,  approximately,  when  edc  is  a  right 
angle. 

Let  0,,  0,  be  the  corresponding  values  of  0  and  0,  and  let 


262  HYDRA  ULICS. 

yt  be  the  angle  between  cd  and  the  tangent  at  d  to  the  wheel's 
periphery.     Then 


and 


=  90°  - 


sn  r,  ._.    g 
sin  r 


two  equations  giving  0,  and  0^ 

Also,  if  ^  is  drawn  perpendicular  to  od, 

de      r  —  R  cos 


tan  y  =  cot  <:#  <?  =  —  = 


ce  R  sin  fi 

The  vertical  distance  between  the  points  where  spilling  be- 
gins and  ends,  viz.,  rl  (sin  0l  —  sin  0)  can  now  be  determined. 
The  pitch-angle(=  rp)  is  the  angle  between  two  consecutive 

buckets  so  that  ^  =          .     In   order  to  obtain  a  small  angle 

(=:  y^  between  the  lip  of  the  bucket  and  the  wheel's  periphery, 
it  is  usual  to  make  the  bucket  angle  ft  greater  than  if}. 
For  example, 

5         5  360°      450° 


The  interval  between  the  buckets  should  be  at  least  suf- 
ficient to  prevent  any  bucket  dipping  into  the  one  below  at  the 
moment  the  latter  begins  to  spill. 

Let  coo'.  Fig.  158,  be  the  division  angle  and  t  the  thickness 
of  the  bucket. 

Then 


approximately,  and  therefore 

(3) 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS.  263 
Also,  by  equation  2, 

,   2nQ 


„    r>  „.  9  Jt 

^Sb*.^^ 

Z,  £  £ 


..     (4) 

W 


These  two  last  equations  give  N  and  0. 
The  number  of  buckets  may  also  be  approximately  found 
from  the  formula 


In  practice  the  bucket  may  be  delineated  as  follows : 
Let  ddr  =  distance  between  two  buckets. 

56  d 

Take  dd"  =  ~ dd'  to  -  dd'\  also  take  fo  =  -,  and  join  dc. 

This  gives  the  form  of  a  suitable  wooden  bucket. 


FIG.  158. 

If  the  bucket  is  of  iron,  a  circular  arc  is  substituted  for  the 
portions  be,  cd. 

Again,  let/w,  Fig.  159,  be  the  thickness  of  the  stream  just 
before  entering  the  bucket. 

Let  dn  be  the  thickness  of  the  stream  just  after  entering 
the  bucket. 

Let  \  be  the  angle  between  the  bucket's  lip  and  the  wheel's 
periphery. 


264  HYDRA  ULICS. 

Then 

mbdul  —  capacity  of  bucket  =  bv^  . pm  =  bV. dn 

=  bv^dp  sin  y  =  b  V.  dp .  sin  A, 

and  therefore 


~  v.smr"  FsinA' 
Now  overshot  wheels  cannot  be  ventilated,  and  it  is  conse- 


FIG.  159. 

quently  necessary  to  leave  ample  space  above  the  entering 
stream  for  the  free  exit  of  air.  Thus,  neglecting  float  thick- 
ness, 

'  =  distance  between  consecutive  floats 


=  <W'(Fig.  158)  >  <//  > 
and  N,  the  number  of  buckets, 

2  Try,  F  sin  \ 
mdu, 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  26$ 


For  efficient  action  the  number  of  the  buckets  is  much  less 
than  the  limit  given  by  this  relation,  often  not  exceeding  one 
half  of  such  limit. 

If  y  is  very  small,  V=vl  —  u^  approximately,  and  therefore 


The  capacity  of  a  bucket  depends  upon  its  form  ;  and  the 
bucket  must  be  so  designed  that  the  water  can  enter  freely 
and  without  shock,  is  retained  to  the  lowest  possible  point,  and 
is  finally  discharged  without  let  or  hindrance.  Hence  flat 
buckets,  Fig.  160,  are  not  so  efficient  as  the  curved  iron  bucket 
in  Fig.  163  and  as  the  compound  bucket  made  of  three  or  two 


FIG.  1 60. 


FIG.  161. 


FIG.  162. 


FIG.  163. 


FIG.  164. 


pieces  in  Figs.  161,  162,  164.  The  resistance  to  entrance  is 
least  in  the  curved  bucket,  as  there  are  no  abrupt  changes  of 
direction  due  to  angles.  The  capacity  of  a  compound  bucket 
may  be  increased,  without  diminishing  the  ease  of  entrance,  by 
making  the  inner  portion  strike  the  inner  periphery  at  an 


266 


HYDRA  ULICS. 


acute  angle,  Fig.  164.  The  objection  to  this  construction, 
especially  if  the  relative  velocity  V  is  large,  is  that  the  water 
tends  to  return  in  the  opposite  direction  and  escape  from  the 
bucket. 

Let  bed,  efg,  Fig.  165,  represent  two  consecutive  buckets  of 
an  overshot  wheel  turning  in  the  direction  shown  by  the  arrow. 


FIG.  165. 


Water  will  cease  to  enter  the  bucket-space  between 
efg,  and  impact  will  therefore  cease,  when  the  upper  parabolic 
boundary  of  the  supply-stream  intersects  the  edge  b.  The  last 
fluid  elements  will  then  strike  the  water  already  in  the  bucket 
at  a  point  M,  whose  vertical  distance  below  b  may  be  desig- 
nated by  z.  The  velocity  v'  with  which  the  entering  particles 
reach  M  is  given  by  the  equation 


(0 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  267 

Again,  while  the  fluid  particles  move  from  b  to  M  let  the 
buckets  move  into  the  positions  b'c'd' ,  e'f'g'. 
Let  arc  bb'  =  s1  =  eer. 
Let  arc  bM  =  st. 

Let  T  be  the  time  of  movement  from  b  to  b'  (or  b  to  M\ 
Then 


s. •  =  uT 


and 


assuming  that  the  mean  velocity  from  b  to  M  is  an  arithmetic 
mean  between  the  initial  and  final  velocity  of  entrance.     Thus 


l  -f-  ^i 


Also,  since  the  angle  between  bM  and  the  wheel's  periphery 
is  small,  it  may  be  assumed  that 

the  arc  bM '  =  be  -\-  ef-\-  ee'y  approximately, 
27tr, 


N          N         u 


, 
+**' 


/,T  r^7  ,Vi—    U          27Cri     Vi   —    U  \ 

(Note.—ef—  eb  —  =  eb-  -  =  -^T.  -  -  ,  nearly.) 
\  J  u  u  N          u  J  i 


Thus 


and  by  equations  2  and  3, 

(vi  +  v*'  —  2U\  _  27tri  !!L 
S\         2u          I  ~    N    u> 


268 


HYDRA  ULICS. 


an  equation  giving  approximately  the  distance  sl  passed 
through  by  a  float  during  impact.  The  buckets  can  now  be 
plotted  in  the  positions  they  occupy  at  the  end  of  the  impact. 
The  amount  of  water  in  each  bucket  being  also  known,  the 
water-surface  can  be  delineated,  and  hence  the  vertical  distance 
x  can  be  at  once  found. 

20.  Useful  Effect  —  (a)  Effect  of  Weight.  —  The  wheel 
should  hang  freely,  or  just  clear  the  tail-water  surface,  and 
the  total  fall  is  measured  from  the  surface  of  the  water  in  the 
tail-race  to  the  water-surface  just  in  front  of  the  sluices  through 
which  the  water  is  brought  on  to  the  wheel. 


FIG.  1 66. 

Let  hlt  Fig.  166,  be  the  vertical  distance  between  the  cen- 
tres of  gravity  of  the  water-areas  of  the  first  and  last  buckets 
before  spilling  commences.  Then 

//,  =  R  cos  d  -\-  rl  sin  0,  very  nearly. 

Let  h^  be  the  vertical  distance  between  the  centres  of 
gravity  of  the  water-area  of  the  bucket  which  first  begins  to 
spill,  and  the  point  at  which  the  spilling  is  completed.  Then 

h^  —  r,(sin  0,  —  sin  0),  very  nearly. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  269 


The  useful  work  per  sec.  =  '^Q(hl  +  kh^  k  being  a  frac- 
tion <  I  and  approximately  =  .5. 

Let  A0  be  the  water-area  in  the  bucket  which  first  begins 
to  spill. 

Between  this  bucket  and  the  one  which  is  first  emptied, 
i.e.,  in  the  vertical  distance  /z2  ,  insert  an  even  number  s  of 
buckets,  and  let  their  water-areas  Al  ,  A9,  A3  ,  .  .  .  As  be  care- 
fully calculated. 

Let  Qm  be  the  mean  amount  of  water  per  bucket  in  the 
discharging  arc. 

Let  Am  be  the  mean  water-area  per  bucket  in  the  discharg- 
ing arc. 

Then 


The  value  of  k  can  now  be  easily  found,  since 


Qm_Am 

~-~" 


Let  q  be  the  varying  amount  of  water  in  a  bucket  frorrr 
which  spilling  is  taking  place,  and  at  any  moment  let  y  be  the 
vertical  distance  between  the  outer  edge  of  the  bucket  and  the 
surface  of  the  water  in  the  tail-race. 

q  is  a  function  of  y  and  depends  upon  the  contour  of  the 
water  in  the  bucket. 

Let  Y  be  the  mean  value  of  y  between  the  points  where 
spilling  begins  and  ends,  i.e.,  for  values^,  and  j/a  of  y.  Then 


y\ 
since 


Jy  .dq=yq  —  Jq .  dy. 


2/O  HYDRA  ULICS. 

Again,  the  elementary  quantity  of  water,  dq,  having  an 
initial  velocity  equal  to  that  of  the  wheel,  viz.,  &,  falls  a  dis- 
tance y  and  acquires  a  velocity  =  <J  u'  -\-  2gy. 

Thus    it     flows    away    in    the   tail-race    causing   a   loss   of 

w  .dq  (  if 

energy  =  -"(*  +  2^7)  =  w  • 


Hence  the  total  loss  of  energy  between  the  points  where 
spilling  begins  and  ends 


Overshot  and  pitch-back  wheels  do  not  work  well  in  back- 
water, as  they  lift  a  greater  or  less  weight  of  water  in  rising 
above  the  surface. 

If  the  water-level  in  the  race  is  liable  to  variation  it  is  better 
to  diminish  the  diameter  of  the  wheel  and  design  it  so  that  it 
may  never  be  immersed  to  a  greater  depth  than  12  inches. 

(b)  Effect  of  Impact.  —  The  head  h'  required  to  produce  the 
velocity  v  with  which  the  water  reaches  the  wheel  is  theoret- 

v* 
ically  —  —  ;  but  as  there  is  a  loss  of  at  least  5  per  cent  in  the 

o 

most  perfect  delivery,  it  is  usual  to  take  h'  =  v-^,  an  average 

o 

value  of  v  being  I.I. 

Let  the  water  enter  the  bucket  in  the  direction  ac,  Fig. 
167.  Take  ac  =  vr  The  water  now  moves  round  with  a 
velocity  u  (assumed  the  same  as  that  of  the  division  circle), 
and  leaves  the  wheel  with  the  same  velocity.  Take  ab  in  the 
direction  of  the  tangent  to  the  division  circle  at  the  point  of 
entrance  =  u.  The  component  be  represents  the  relative 
velocity  V  of  the  water  with  respect  to  the  bucket,  and  this 
velocity  is  wholly  destroyed,  ab  must  necessarily  be  parallel  to 
the  outer  arm  of  the  bucket,  so  that  there  may  be  no  loss  of 
shock  at  entrance.  Then  the  impulsive  effect 


g 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  2?  I 

But 

V*  =  v?  -\-  if  —  2v \u  cos  y, 

y  being  the  angle  through  which  the  water  is  deviated  from 
its  original  direction  at  the  point  of  entrance. 


FIG.  167. 
Hence  the  impulsive  effect 


wQ 
= u(v^  cos  y  —  u), 

o 


and  the  TOTAL  USEFUL  EFFECT 


i+^2)+  ^M7'i  cos  K—  &)—'loss  due  to  journal  friction, 

o 


The  loss  due  to  journal  friction 


p  being  the  radius  of  the  axle  and  Wthe  weight  of  the  wheel. 


2/2  HYDRA  ULICS. 

21.  A  pitch-back  or  high  breast  wheel  is  to  be  preferred 
to  an  overshot  wheel  when  the  surface-levels  of  the  head-  and 
tail-water  are  liable  to  very  considerable  variation. 

In  the  pitch-back  wheel  the  water  is  admitted  by  an  adjust- 
able sluice  into  the  buckets  on  the  same  side  as  the  supply- 
channel,  Fig.  168.  Thus  the  wheel  revolves  in  the  direction 


FIG.  168. 

in  which  the  water  leaves,  and  the  drowning  of  the  wheel  is 
prevented.  Further,  the  buckets  may  be  now  ventilated,  Fig. 
169,  and  may  therefore  be  placed  closer  together  than  in  the 
unventilated  overshot  wheel. 

The  efficiency  of  the  pitch-back  is  at  least  equal  to  that  of 
the  overshot. 

22.  The  Jet  Reaction  Wheel  (Scotch  Turbine). — In  this 
form  of  motor  the  water  enters  the  centre  of  the  wheel,  spreads 
out  radially  in  tubular  passages,  and  issues  from  openings  at 
the  ends  tangentially  to  the  direction  of  rotation. 

Fig.  170  represent  the  simplest  wheel  of  this  class.  In  Eng- 
land it  is  known  as  Barker's  mill,  and  in  Germany  it  is  called 
Segner's  water-wheel. 

A  reaction  wheel  may  have  several  tubular  passages,  as  in 
Fig.  172,  and  the  vertical  chamber  XY  may  be  cylindrical, 
rectangular,  or  conical. 

Let  r  be  the  horizontal  distance  between  the  axis  of  aa 
orifice  and  the  axis  of  the  vertical  chamber. 

Let  h  be  the  head  of  water  over  the  orifices  when  closed. 

Let  v  be  the  velocity  of  erflux  relatively  to  the  tube  when 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  2?$ 

the   orifices  are   open,  and   let  Fbe  the  corresponding  linear 
velocity  of  rotation  at  the  centre  of  an  orifice.     Then 


cv  being  the  coefficient  of  velocity. 

FIG.  1 70. 


FIG.  171. 


2  74  HYD  RA  ULICS. 

The  absolute  velocity  of  efflux  '=  v  —  V. 

v-V 
The  angular  momentum  of  each  pound  of  water  = r. 

o 

The  useful  work  of  each  pound  of  water 

v-V  V       V. 

= r—  =  —(v  —  V\ 

g       r       g 

The  total  work  of  each  pound  of  water  =  h. 


FIG.  172. 


The  efficiency 

useful  work  _ 
Total  work 


-  V} 


=  ^  suppose> 


useful  work  _  v  —  V 

The  reaction  =  linear  veiocity  of  rotation  =        g 

For  a  maximum  efficiency 


=  o  = 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   2?$ 

Hence 

«f —  2Z/F+ *.•?"  =  o, 
and  therefore 


v  =  V(i  +  Vi  -  c,')  .......    (4) 

Experience  indicates  that  the  greatest  efficiency  corresponds 
to  a  speed  of  rotation  equal  to  the  velocity  due  to  a  head  h, 
i.e.,  to  a  value  of  V  given  by 


.  -  ,    .....    (5) 

By  equations  (i)  and  (5) 

«f  =  4V**   .......     (6) 

and  therefore,  by  equations  (4),  (5),  and  (6), 

o 

c*  =  ~»     or     c,  =  .94  .......    (7) 

Hence,  by  equations  (3),  (5),  (6),  and  (7), 

the  maximum  efficiency  =  —  . 

o 

Thus  one  third  of  the  head  is  lost,  and  of  this  amount  the 

(v—  F)2/      h\ 
portion  ---  —  —  ^=  -j  is  carried  away  by  the  effluent  water. 

The  portion  -  --  (=  -kj  is  lost  in  frictional  resistance,  etc. 
Again, 


=  —jt  |  cjj*  +  -yT—  terms  cont'g  higher  powers  of  -~\  —  i  |  . 


276  HYDRA  ULICS. 

The  efficiency  therefore  increases  with  F,  but  the  value  of 
V  is   limited  by    the   practical   consideration    that,    even   at 

moderately  high  speeds,  so  much  of 
the  head  is  absorbed  by  friction  as 
to  sensibly  diminish  the  efficiency. 

The  serious  practical  defects  of 
this  wheel  are  that  its  speed  is  most 
unstable  and  that  it  admits  of  no 
efficient  system  of  regulation  for  a 
varying  supply  of  water. 

The   Scotch   or   Whitelaw's   tur- 
J73.  bine,    Fig.     173,     excepting    in   the 

curved  arms,  does  not    differ    essentially   from  the   reaction 
wheel  just  considered. 

23.  Reaction  and  Impulse  Turbines.— All  turbines  be- 
long to  one  of  two  classes,  viz.,  Reaction  Turbines  and  Impulse 
Turbines,  and  are  designed  to  utilize  more  or  less  of  the  avail- 
able energy  of  a  moving  mass  of  water. 

In  a  reaction  turbine  a  portion  of  the  available  energy  is 
converted  into  kinetic  energy  at  the  inlet  surface  of  the  wheel. 
The  water  enters  the  wheel-passages  formed  by  suitably 
curved  vanes,  and  acts  upon  these  vanes  by  pressure,  causing 
the  wheel  to  rotate.  The  proportions  of  the  turbine  are  such 
that  there  is  a  particular  pressure  (hence  the  term  pressure- 
turbine)  at  the  inlet  surface  corresponding  to  the  best  normal 
condition  of  working.  Any  variation  from  this  pressure, 
caused,  e.g.,  by  the  partial  closure  of  the  passages  through 
which  the  water  passes  to  the  wheel,  changes  the  working  con- 
ditions and  diminishes  the  efficiency.  In  order  to  avoid  such 
a  variation  of  pressure,  it  is  essential  that  there  should  be  a 
continuity  of  flow  in  every  part  of  the  turbine  ;  the  wheel- 
passages  should  be  kept  completely  filled  with  water,  and 
therefore  must  receive  the  water  simultaneously;  Such 
turbines  are  said  to  have  complete  admission.  The  admission 
is  partial  when  the  water  is  received  over  a  portion  of  the  inlet 
surface  only. 

In   an  impulse  (Girard)  turbine,  Figs.  174,  175,  the  energy 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS. 

of  the  water  is  wholly  converted  into  kinetic  energy  at  the 
inlet  surface.  Thus  the  water  enters  the  wheel  with  a  velocity 
due  to  the  total  available  head  and  therefore  without  pressure, 
is  received  upon  the  curved  vanes,  and  imparts  to  the  wheel 
the  whole  of  its  energy  by  means  of  the  impulse  due  to  the 


FIG.  174.  FIG.  175. 

Girard  Turbine  for  Low  Falls.  Girard  Turbine  for  High  Falls. 

gradual  change  of  momentum.  Care  must  be  taken  to  ensure 
that  the  water  may  be  freely  deviated  on  the  curved  vanes, 
and  hence  such  turbines  are  sometimes  called  turbines  with  free 
deviation.  For  this  reason  the  water-passages  should  never  be 
completely  filled,  and  the  water  should  flow  through  under  a 
pressure  which  remains  constant.  In  order  to  ensure  an  un- 
broken flow  through  the  wheel-passages  and  that  no  eddies 
are  formed  at  the  backs  of  the  vanes,  ventilating  holes  are 
arranged  in  the  wheel  sides,  Fig.  177.  Figs.  176  and  177  also 
show  the  relative  path  AB  and  the  absolute  path  CD  traversed 
by  the  water  in  an  inward-flow  and  a  downward-flow  turbine. 
If  there  is  a  sufficient  head,  the  wheel  may  be  placed  clear 


278 


HYDRAULICS. 


above  the  tail-water,  when  the  stream  will  be  at  all  times  under 
atmospheric  pressure.    With  low  falls  the  wheel  may  be  placed 

in  a  casing  supplied  with  air  from 
an  air-pump  by  which  the  surface 
of  the  water   may  be  kept  at  an 
invariable    level   below   the  outlet 
orifices,  which  is  essential  for  per- 
fectly  free   deviation.     While  the 
wheel-passages  of  a  reaction   tur- 
bine  should    be   kept    completely 
'filled  with  water,  no  such  restric- 
tion is  necessary  with  an  impulse 
turbine.     The  supply  may  be  par- 
tially  checked    and    the   water    may   be    received    by  one  or 
more  vanes  without  affecting  the  efficiency.  '  Thus  the  dimen- 
sions of  an  impulse   turbine   may   vary   between    very    wide 


TAIL  WATER 


FIG.  177. 

limits,  so  that  for  high  falls  with  a  small  supply,  a  compara- 
tively large  wheel  with  low  speed  may  be  employed.  The 
speed  of  a  reaction  turbine  under  similar  conditions  would  be 
disadvantageously  great,  and  any  considerable  increase  of  the 
diameter  would  largely  increase  the  fluid  friction  and  would 
also  render  the  proper  proportioning  of  the  vane-angles 
almost  impracticable.  Impulse  turbines  may  have  complete 
or  partial  admission,  while  in  reaction  turbines  the  admission 
should  be  always  complete,  as  in  Fig.  178,  which  shows  the 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS. 

relative  path  AB  and  absolute  path  CD  traversed  by  the  water. 
When  there  is  an  ample  supply  of  water  the  reaction  turbine 
is  usually  to  be  preferred,  but  on  very  high  falls  its  speed 


FIG.  178. 

becomes  inconveniently  great  and  it  is  then  better  to  adopt  a 
turbine  of  the  impulse  type.  The  diameter  of  the  wheel  can 
then  be  increased  and  the  speed  proportionately  diminished. 

The  Hurdy-gurdy  is  the  name  popularly  given  to  an 
impulse  wheel  which  was  introduced  into  the  mining  districts 
of  California  about  the  year  1865.  Around  the  periphery  of 
the  wheel  is  arranged  a  series  of  flat  iron  buckets,  about 
4  to  6  in.  in  width,  which  are  struck  normally  by  a  jet  of 
water  often  not  more  than  three  eighths  of  an  inch  in 
diameter.  Theoretically,  the  efficiency  of  such  an  arrange- 
ment cannot  exceed  50  per  cent  (Art.  7),  while  in  prac- 
tice it  rarely  reaches  40  per  cent.  The  best  speed  of  the 
wheel,  in  accordance  with  both  theory  and  practice,  is  one 
half  of  that  of  the  jet.  Although  the  efficiency  is  so 
low,  the  wheel  found  great  favor  for  many  reasons.  Any 
required  speed  could  be  obtained  by  a  suitable  choice  of 
diameter ;  the  plane  of  the  wheel  could  be  placed  in  any 
convenient  position ;  the  wheel  could  be  cheaply  constructed 
and  was  largely  free  from  liability  to  accident.  Hence  it  was 
of  the  utmost  importance  to  increase,  if  possible,  the  efficiency 
of  a  wheel  possessing  such  advantages.  Obviously  a  first  step 
was  to  substitute  cups  for  the  flat  buckets,  the  immediate 
result  necessarily  being  a  very  large  increase  in  the  efficiency. 
This  was  increased  still  further  by  the  adoption  of  double 


2  80  H  YDRA  ULICS. 

buckets,  Fig.  179,  that  is,  curved  buckets  divided  in  the  middle 
so  that  the  water  is  equally  deflected  on  both  sides. 

Thus  developed,  the  wheel  is  widely  and  most  favorably 
known  as  the  Pelton  wheel,  Fig.  179.  Its  efficiency  is  at  least 
80  per  cent,  and  it  is  claimed  that  it  often  rises  above  90  per 
cent.  The  power  of  the  wheel  does  not  depend  upon  its 
diameter,  but  upon  the  available  quantity  and  head  of  water. 
The  water  passes  to  the  wheel  through  one  or  more  nozzles, 


FIG.  179. 

having  tips  bored  to  suit  any  required  delivery.  These  tips 
are  screwed  into  the  nozzles  and  can  be  easily  and  rapidly 
replaced  by  others  of  larger  or  smaller  size,  so  that  the  Pelton 
is  especially  well  adapted  for  a  varying  supply  of  water.  It  is 
claimed  that  in  this  manner  the  power  may  be  varied  from  a 
maximum  down  to  25  per  cent  of  the  same  without  appreci- 
able loss  of  efficiency. 

The  character  of  the  construction  of  turbines  has  led  to 
their  being  classified  as  (i)  Radial-flow  turbines;  (2)  Axial- 
flow  turbines  ;  (3)  Mixed-flow  turbines. 

In  Radial-flow  turbines  the  water  flows  through  the  wheel 
in  a  direction  at  right  angles  to  the  axis  of  rotation  and 
approximately  radial.  The  two  special  types  of  this  class  are 
the  Outward-flow  turbine,  invented  by  Fourneyron,  and  the 
Inward-flow  or  Vortex  turbine,  invented  by  James  Thomson. 
In  the  former,  Figs.  180  and  181,  the  water  enters  a  cylindrical 
chamber  and  is  led  by  means  of  fixed  guide-blades  outwards 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  28 1 

from  the  axis.     It  is  distributed  over  the  inlet-surface,  passes 
through  the  curved  passages  of  an  annular  wheel  closely  sur- 

FIG.  1 80. 


FIG.  181. 


rounding  the  chamber,  and  is  finally  discharged  at  the  outer 
surface.     The  wheel  works  best  when  it  is   placed  clear  above 


282  HYDRA  ULICS. 

the  tail-water.  A  serious  practical  defect  is  the  difficulty  of 
constructing  a  suitable  sluice  for  regulating  the  supply  over 
the  inlet-surface.  Fourneyron  was  led  to  the  design  of  this 
turbine  by  observing  the  excessive  loss  of  energy  in  the  ordi- 
nary Scotch  turbine,  or  reaction  wheel,  and  introduced  guide- 
blades  in  order  to  give  the  water  an  initial  forward  velocity 
and  thus  cause  a  diminution  of  the  velocity  of  the  water  leav- 
ing the  outlet-surface. 

In  the  Inward-flow  or  Vortex  turbine,  Figs.  182  and  183, 
the  wheel  is  enclosed  in  an  annular  space,  into  which  the 
water  flows  through  one  or  more  pipes,  and  is  usually  dis- 
tributed over  the  inlet-surface  of  the  wheel  by  means  of  four 
guide-blades.  The  water  enters  the  wheel,  flows  towards  the 
space  around  the  axis,  and  is  there  discharged.  This  turbine 
possesses  the  great  advantage  that  there  is  ample  space  outside 
the  wheel  for  a  perfect  system  of  regulating-sluices. 

Axial- flow  turbines,  Figs.  184,  are  also  known  as  Parallel 
and  Downward-flow  turbines  and  are  sometimes  called  by  the 
names  of  the  inventors,  Jonval  and  Fontaine.  In  these  the 
water  passes  downward  through  an  annular  casing  in  a  direction 
parallel  to  the  axis  of  rotation,  and  is  distributed  by  means  of 
guide-blades  over  the  inlet-surface  of  an  adjacent  wheel.  It 
enters  the  wheel-passages  and  is  finally  discharged  vertically,  or 
nearly  so,  at  the  outlet-surface.  The  sluice  regulations  are 
worse  even  than  in  the  case  of  an  outward-flow  turbine,  but 
there  is  this  advantage,  that  the  turbine  may  be  placed  either 
below  the  tail-water,  or,  if  supplied  with  a  suction-pipe,  at  any 
point  not  exceeding  30  ft.  above  the  tail-water. 

If  a  turbine  is  designed  so  that  the  pressure  at  the  clear- 
ance between  the  casing  and  the  wheel  is  nil,  and  with  curved 
passages  in  the  form  of  a  freely  deviated  stream,  it  becomes 
what  is  called  a  Limit  turbine.  In  its  normal  condition  of 
working  it  is  an  Impulse  turbine,  but  when  drowned,  it  is  a 
Reaction  turbine,  with  a  small  pressure  at  the  clearance.  For 
moderate  falls  with  a  varying  supply  its  average  efficiency  is 
higher  than  that  of  a  pressure  turbine. 

The  Mixed-  or  Combined-flow  (Schiele)  turbine  is  a  combi- 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   283 

nation  of  the  radial  and  axial  types.     The  water  enters  in  a 
nearly  radial  direction  and  leaves  in  a  direction  approximately 


FIG.  182. 


X  — 


///////////////^^^^^ 


1 


--T 


FIG.  183. 

parallel  to  the  axis  of  rotation.     This  type  of  turbine  admits 
of  a  good  mode  of  regulation  and  is  cheap  to  construct. 

24.  Theory  of  Turbines  (Figs.  185  to  188).— Denote  in- 


284 


HYDRA  ULICS. 


ward-flow,  outward-flow,  and  axial-flow  turbines  by  I.  F.,  O.  F., 
and  A.  F.,  respectively. 


FIG.  184. 

Let  r,,  ra  be  the  radii  of  the  wheel  inlet  and  outlet  surfaces 
or  an  I.  F.  or  O.  F. 

Let  rlt  rt  be  the  outer  and  inner  radii  of  the  wheel  inlet- 
surface  of  an  A.  F. 

Let  R  be  the  mean  radius  \==  r*  "^  r*J  of  an  A.  F.,  assumed 
constant  throughout. 


FIG  185. — Section  of  an  inward-flow  turbine. 

Let  Alf  A,  be  the  areas   of  the  wheel  inlet  and   outlet 
orifices. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    285 


FIG.  186. — Enlarged  portion  of  the  section  through  XY,  Fig.  185. 


FIG.  187. — Enlarged  portion  of  a  section  through  XY,  Fig.  180,  of  an  outward- 
flow  turbine. 


FIG.  188. — Enlarged  portion  of  a  cylindrical  section  XY,  Fig.  184,  of  a  down- 
ward-flow turbine  developed  in  the  plane  of  the  paper. 


286  HYDRAULICS. 

Let  dlt  dt  be  the  depths  of  the  same  in  an  I.  F.  or  O.  F. 
Let  dlt  d^  be  the  widths  of  the  same  in  an  A.  F. 
Let  h  be  the  thickness  of  the  wheel  in  an  A.  F. 
Let  Hl  be  the  effective  head  over  the  inlet-surface  of  the 
wheel.     This  is  the  total  head  over  the  inlet- 
surface  diminished  by  the  head  consumed  in 
frictional  resistance  in  the  supply-channel,  and 
by  the  head  lost  in  bends,  sudden  changes  of 
section,  etc. 

Let  HI  be  the  fall  from  the  outlet-surface  to  the  surface  of 
the  water  in  the  tail-race.     If  the  turbine  is 
submerged,  then  H9  is  negative. 
Let  vlt  vt  be  the   absolute  velocities  of  the  water  at  the 

inlet-  and  outlet-surfaces. 

Let  ult  #,  be  the  absolute  velocities  of  the  inlet-  and  outlet- 
surfaces. 
Let  V^  Vi  be  the  velocities  of  the  water  relatively  to  the 

wheel,  at  the  inlet-  and  outlet-surfaces. 
Let  GO  be  the  angular  velocity  of  the  wheel. 
Let  the  water  enter  the  wheel  in  the  direction  act  making 
an  angle  y  with  the  tangent  ad.     Take  ac  to  represent  vl  and 
ad  to  represent  ult     Complete  the  parallelogram  bd.    The  side 
ab  represents  Vlt  and  in  order  that  there  may  be  no  shock  at 
entrance,  ab  must  be  tangential  to  the  vane  at  a.     Again,  at/ 
drawy^-,  a  tangent  to  the  vane,  and//£,  a  tangent  to  the  wheel's 
periphery. 

Take  fg  and  fk  to  represent  V^  and  u^  respectively.  Com- 
plete the  parallelogram  gk.  The  diagonal /$  must  represent 
in  direction  and  magnitude  the  absolute  velocity  v^  with  which 
the  water  leaves  the  wheel.  Let  the  angle  hfk  =  d. 

The  tangential  component  of  the  velocity  of  the  water  as 
it  enters  or  leaves  the  wheel  is  termed  the  velocity  of  whirl, 
and  the  radial  component  the  velocity  of  flow.  Denote  these 
components  respectively  by 

vj,  vr'  at  the  inlet-surface  ; 
v»' i  vr"  at  the  outlet-surface. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  28/ 

Let  the  angle  bad  =  1 80°  —  a, 
Let  the  angle  gfk  =  180°  —  ft. 
Draw  cm  perpendicular  to  ad,  and  hn  to  fk. 

Then  at  the  inlet-surface, 

vj=.  v^  cos  y  —  ac  cos  y  =  #;#  =  ad^dm  =  #, —  P,  cos  a  ;     (i) 
?>/  =  z/,  sin  ^  —  £;#  =  Vl  sin  or ; (2) 

and  at  the  outlet-surface 

vj'  =  z>a  cos  6  =fn  =.fk  ±kn  =  u9—  V^  cos  /? ;    .     (3) 
vr"  =  ^2  sin  6  =  /*«  —  F2  sin  /? (4) 

Let  g  be  the  volume  of  water  passed  per  second.     Then    • 


in  an  I.  F.  or  O.  F. 

Vr'Ai   =  VrZTtridi  •=  Q 


(5) 


in  an  A.  F. 


i    =   Q 


(5) 


In  equations  (5)  the  thickness  of  the  vanes  has  been  disre- 

garded.    If  0  is  the  angle  between  the  vane,  of  thickness  BC, 

A/  and   the   wheel's    periphery  AB,  then   the  space 

^f^j      occupied  by  the  vane  along  the  wheel's  periph- 

/  /       ery  is  AB  =  BC  cosec  0. 

/  Let  n  be  the  number  of  the  guide-vanes  and  / 

FlG-  I89-  their  thickness. 

Let  #,  be  the  number  of  the  wheel-vanes  and  /,  ,  /2  their 
thickness  at  the  inlet-  and  outlet-surfaces,  respect- 
ively. 
Then,  in  a  radial-flow  turbine, 


Al  —  -fadl\2nrl  —  nt  cosec  y  —  nvtl  cosec  a\  .     .     (6) 
and 

^.  =  TWi2^.-  *i**  cosec  ft\>     ......     (7) 

T9¥  being  a  fraction  depending  on  practical  considerations. 


288 


HYDRAULICS. 


In  an  axial-flow  turbine  R  is  to  be  substituted  for  rl  ind  ry 
in  the  values  of  Al  and  A9. 

nl  may  be  made  equal  to  n  -f-  I  or  n  -f-  2. 

Again,  as  the  water  flows  through  the  wheel  its  angular 
momentum  relatively  to  the  axis  of  rotation  is  changed  from 

— rjsj  at  the  inlet-  to  rj)J'  at  the  outlet-surface. 

o  o 

Hence,  if  T  is  the  effective  work  done  by  the  water  on  the 
turbine,  and  GO  the  angular  velocity  of  the  turbine, 


in  an  I.  F.  or  O.  F. 

in  an  A.  F. 

T  -  ^(vv'n  -  vw"rt)<o 

g 

wQ 
g 

s 

g 

a/')«i»    •     •    (8> 

since 

since 

Ui             «2 

r*=  —  =  °°>    -   -  (9) 

'1              *  2 

and  the  hydraulic  efficiency 

T          vw'ui  -  z>«"wa 

and  the  hydraulic 

r/        f 
\Viv 

efficiency 
-«^")«i     /I0x 

wQH,                gff,          '   ( 

wQ(H,  +  A)         g(i 

yi  +  A)   ' 

Equation   10  is  the  fundamental  equation  upon  which  the 
whole  design  of  turbines  depends. 
From  the  triangle  abc, 


V*  =  v*  +  u*  —  2vlul  cos  y,     .     .     .     .    (ii) 


and 


sn  y 
sin  of 


(12) 


From  the  triangle  ./M, 

»,1  =  «,1+F,1-2«,r,COS/» (I3) 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   289 


Again,  if  —  ,  --  are  the  pressure-heads  at  the  inlet-  and  outet- 

w  w 


surfaces  of  the  wheel  of  a  REACTION  TURBINE, 

A -A 


—  =ff.  - 


w 


(14) 


In  an  IMPULSE  TURBINE  the  water  is  under  atmospheric 
pressure  only,  and  therefore 


05) 


To  make  allowance  for  hydraulic  resistances  k^—  may  be 

o 

v* 
substituted  for  —  in  equations  14  and  15,  a  mean  value  of  kl 

o 

^  '      I0 

being  —  . 

'  9 

Applying  Bernoulli's  theorem  to  the  filament  from  a  to/, 

%  *  _  u  » 
and  taking  account  of  the  head  -  -  due  to  centrifugal 

force  — 


In  a  reaction  I.  F.  or  O.  F. 


Wt         2g        W 

and  therefore 
VJ-  V?  _/i 


In  words,  the  change  of  en- 
ergy from  atof  =  work  due 
to  pressure  -|-  work  due  to 
centrifugal  force. 

In  an  impulse  I.  F.  or  O.  F. 
V^~  r'*  =  ***~"1*.     (18) 


In  a  reaction  A.  F. 


2g 


and  therefore 


In  words,  the  change  of  en- 
ergy from  a  to  f  =  work  due 
to  pressure  -f-  work  due  to 
gravity.  The  work  due  to  A 
centrifugal  force  is  evidently 
nil. 

In  an  impulse  A.  F. 

^  ~  V^  =  h-    -   •  (I8> 


290 


HYDRA  ULICS. 


To  make  allowance  for  hydraulic  resistances  £,  F,2  may  be 
substituted  for  V9  in  equations  17  and  18,  a  mean  value  of  £a 
being  i.i. 

For  a  maximum  effect  the  water  should  leave  the  wheel 
without  velocity,  i.e.,  vt  should  be  nil.  But  this  value  of  v^  is 
impracticable,  as  no  water  could  then  pass  through  the  wheel. 
It  is  usual  either  to  make  the  velocity  of  whirl  (vm")  at  the 
outlet-surface  equal  to  nil,  or  to  make  the  relative  (F2)  and 
circumferential  (u9)  velocities  at  the  outlet-surface,  equal  and 
opposite.  In  each  case  v9  is  small.  First  let 

»-"  =  <>, d9) 

so  that  the  water  leaves  the  wheel  with  a  much-reduced  ve- 
locity in  a  direction  normal  to  the  out- 
let-surface.    Thus  (Fig.  194),  &\*)**      fy 

£  =  90°;  *.=?*/', 
and 

Aj(j      ^  =  Z>2  COt  /3  =   V9  COS  ft.         (2O) 


\v2-v'rV2 
'     / 


Also,  by  equations  2,  4,  5,  and  20 — 


FIG.  189. 


In  an  I.  F.  or  O.  F. 

~=  vi  sinyridi  =  V*  sin/J>v/2 


211 


•  (21) 


In  an  A.  F. 


=  v\  sin  ydi  =  V*  sin  fid* 


=  «2  tan  fidi.     (21) 


The  following  results  are  now  easily  obtained  : 


In  an  I.  F.  or  O.  F.  : 

Relation  between  the  Vane- 

angles. 

By  equations  9  and  21,  and 
from  the  triangle  acd, 

r\di  sin  y      «3      r*  u\ 


tan 


sin  a. 


In  an  A.  F.  : 

Relation  between  the  Vane- 

angles. 

By  equations  9  and  21,  and 
from  the  triangle  acd, 


d\  sin  y      «2 
</2  tan  @  ~  v 


sin  (a  -\-  y} 
sin  a 


(22) 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.  2QI 


and  therefore 

^-^  cot  ft  =  cot  a  4-  cot  y.    (23) 
TVfla 


and  therefore 

—  cot  ft  =  cot  a  -f-  cot  y.      (23) 
0a 


In  an  I.  F.  or  O.  F.: 

Speed  of  Turbine. 
By  equations  I,  10,  and  19, 


IN  REACTION  TURBINES. 

In  an  A.  F.: 

Speed  of  Turbine. 
By  equations  I,  10,  and  19, 


WQ(HI  —  ^}-  effective  work 

\  ? 


wO  wQ 

~   g  S 

and  therefore 


cosy,    (24) 


Z/2  tt\i>l  ,        x 

Hl —  = cos  y.  .     (25) 

Hence,  by  equations  20,  22, 
and  25, 


COt  /J 


tan  ft  4-  2—  cot^ 

Note. — If  the  water  is  to 
have  no  velocity  of  whirl  (vj) 
relatively  to  the  wheel  at  the 
inlet-surface,  then 

«i  -  vw'  =  o,  .    .    .    (27) 
and  therefore 

a  =  90° 
and 

Vi  Vr 

tan  y  —  —  =  — ,. 

Also,  the  efficiency 


and  thus 


WQ\H1  +  h =-J=  effective  work 

wQ     ,  wQ  .     . 

=—v-wUi  =  — —UM  cos  y,     (24) 
£•  S 

and  therefore 


^i  +  h  -  r1  =  ^~  cos  r- 


Hence,  by  equations  20,  22, 


and  25, 


4-  A)  cot 


-..    (26) 


tan  ft  -\-  i-j-  cot 

. — If  the  water  is  to 
have  no  velocity  of  whirl  (vwf) 
relatively  to  the  wheel  at  the 
inlet-surface,  then 

Ul  -  vw'  =  o,    .    .    (27) 

and  therefore 

a  =  90° 
and 


Also,  the  efficiency 


an    thus 

(28)  uS  =  g(Hi  4- 


.     (28) 


292 


HYDRA  ULICS. 


if  the  efficiency  is  perfect. 
Usually     the    efficiency   of 
good  turbines  is  about  .85. 

Velocity  of  Efflux. 

Z'a2  =  z/a5  tan2  ft 

2,07/1  tan  ft 
(20) 

if  the  efficiency  is  perfect. 
Usually    the    efficiency     of 
good  turbines  is  about  .85. 

Velocity  of  Efflux. 

z/22  =  «22  tan2  ft 
2g(ffi  4-  A)  tan  ft 

tan  ft  -j-  2—  cot  y 

Useful  Work 
2-^-  cot  y 

-wQfft  li  .  (3o) 

tan  /?-(-  2-^-  cot^ 
2—  cot  ^ 

^>/  TT       \      r\                **                                      /       V 

tan  /3-}-  2-f-  cot  y 

Efficiency 

2^  co.r 

</l                       ,    x 

=  wQ(Hi  +  /^)  ^  .    (30) 

tan  ft  -J-  2  —  cot  ^ 
«i 

Efficiency 

2|cotr 

—                       d               '   '     '31) 
tan  />  -J-  2~r  cot  X 

Amount  Q  of  water  passing 
through  turbine 

tan  y#-f-  2-^  cot  v 
«i 

Amount  Q  of  water  passing 
through  turbine 

1        zgVi  tan  ft               . 

,        /ig(Hi  -\-  h}  tan  /5 

-    27rr2</2  A     /  ~                      ^  •    (33) 

y     tan  ^  -(-  2-^  cot  y 

7^^  pressure-head  at  the  in- 

Jff^urfnr.f 

27/ft?i  .    /                       .                 G3> 
y     tan  ft-{-2—co\.y 

The  pressure-head  at  the  in- 
let-surface 

2g 


,„.)       raV,« 

•H»l<  I~ „    OV   9 


tan^ 


2g 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   293 


When  the  turbine  is  work- 
ing freely  in  space  above  the 
surface  of  the  tail-water,  there 
will  be  no  inflow  of  air  if  p^  > 

A»  f-e->  if 


I  >  — o— , 


tan  ft 


'*   sin2;r(tan/?-]-2-2-cot;K) 
d\ 

If  the  turbine  is  drowned 
with  a  head  h'  of  water  over 
the  outlet-surface,  there  will 
be  no  back-flow  of  water  if 


that  is,  if 

7i ^   o 


tan  ft 


When  tne  turbine  is  work- 
ing freely  in  space  above  the 
surface  of  the  tail-water,  there 
will  be  no  inflow  of  air  if  pl  > 

At  Le-'  if 

ffi  <tf_  _  tan  ft  _ 

ffi-4-A       aV2    .   ,  ,,  .    d* 

sin2  ^(tan  p-\-  2—  cot  y) 
d\ 

If  the  turbine  is  drowned 
with  a  head  h'  of  water  over 
the  outlet-surface  there  will 
be  no  back-flow  of  water  if 


$l  -^  ^2  i   j,' 
—  >  --  h  h  , 

w         w 


that  is,  if 


tan  ft 


IN   IMPULSE  TURBINES. 

In  an  I.  F.  or  O.  F.: 

Speed  of  Turbine. 


Since 

V?  =  2gff1  ,       .       .       (35) 

by  equation  22, 

riVi8  sin2  y  _  uj  _  rj    u^ 

and  therefore 


Velocity  of  Efflux. 


=  «2    tan    p 

~nW 


tan  ft  -f-  2-^  cot  y 
d\ 


In  an  A.  F. : 

Speed  of  Turbine. 


Since 


,     -    •    (35) 


by  equation  22, 

dS  sin2  y  _  uf_  _  uf_ 
d<?  tan2  ft  ~  z/i2    ~"  z/xa  ' 

and  therefore 

«22   =  Wj2   =  2gffi    \  ^"a^-  (36) 


Velocity  of  Efflux. 

,2  =  7/22  tan2  ft 


'•       •     (37) 


294 


HYDRA  ULICS. 


Useful  Work 

H       v'\ 
~^j 

Efficiency 


-r^-s{D'r='>-  (39) 


Work 


-/r,g-.in'r).    (38; 
Efficiency 
\—  -ij-  sina  y  =  n.    (39) 


Second,  let 


so  that  the  water  again  leaves  the  wheel  with  a  much-reduced 

velocity.     Evidently  also 


J- 


a  z=  2&2  cos     =  22/a  sn  — 

2 


sn  —  . 

2 


.     (42) 


Also,  by  eqs.  2,  4,  5,  and  42 — 


In  an  I.  F.  or  O.  F. 


Q_ 
zit 


=  «a  sin/?  ra</2.  (43) 


27T 


In  an  A.  F. 

=  »i  si 


=  F3  sin 


.  (43)' 


The  following  results  are  now  easily  obtained  : 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS.  2$$ 

In  an  I.  F.  or  O.  F. : 

Relation  between  the  Vane- 


angles. 

By  eqs.  9  and  43  and  from 
the  triangle  acd 


sin  y  _  w2  _  ^a  «i_ 

sin  /^        z/2       r\    Vi 


r-i  sin  (a  -f*  y) 
ri        sin  a 

and  therefore 

cosec  ^  =  cot 


(44) 


(45) 


Relation  between  the  Vane- 
angles. 

By  eqs.  9  and  43  and  from 
the  triangle  acd 


d\  sin  y  _u<i  _u\ 
di  sin  ft       v\       v\ 

sin  (a  4 


sm 


(44) 


and  therefore 

-i  cosec  ft  =  cot  or  -f  cot  ^.      (45) 


IN  REACTION  TURBINES. 


In  an  I.  F.  or  O.  F.: 

Speed  of  Turbine. 
By  eqs.  14,  17,  and  40 

UiVi  COS  y  =  £-/A  =  UiVw'.        (46) 

Also, 

Wl       sin  (a  +  y} 


Hence, 


sin  a 


+ 


cot  a  tan  X) 
-  tan  ^  cosec  ^.    (47) 


.  —  If  the  velocity  of 
whirl  (^wr)  relatively  to  the 
wheel  at  the  inlet-surface  is  to 
be  nil, 

Ul   —  Vm    =  O,      .      .      (48) 

and  then 


In  an  A.  F.: 

Speed  of  Turbine. 
By  eqs.  14,  17,  and  40 

v\  cos  x  ~=-S(H\  ~T"^)  ==  WiZ'w'-    (46) 

Also, 

«i  _  sin  (a  -}-X) 


sin  a 


Hence 


sin  a  cos 


cot  cr  tan  y) 


+  h~  tan  ^  cosec  /?.  (47) 


TVi?^. — If  the  velocity  of 
whirl  (vj)  relatively  to .  the 
wheel  at  the  inlet-surface  is  to 
be  nil, 

Ul  -  vw'  =  o,    .    .    (48) 

and  then 

f  A).      (49) 


HYDRAULICS. 


Velocity  of  Efflux. 
By  equations  42  and  47 


0   .  0  ft 

82  sin2  - 


i—  tan  L  tan 


Useful  Work 


(50) 


f  i  -  -  tan  6.  tan  A    (51) 
^        d*         2  J 

Efficiency 


Amount  Q  of  Water  passing 
through  Turbine 

=.  2itridivr"  =  27Tra</a  Fa  sin  ft 
=  2itr<id<iU<i  sin  ft 
=  27Tra 


tan  y  sin  /?.    (53) 

Pressure-head  at  Inlet-surface 


by  equations  44  and  47. 

When  the  turbine  is  work- 
ing freely  in  space  there  will 
be  no  inflow  of  air  if  /,  >  /2  , 
i.e.,  if 


When  the  turbine  is 
drowned,  with  a  head  h'  of 
water  over  the  outlet-surface, 


Velocity  of  Efflux. 
By  equations  42  and  47 


ft 

sin2  - 


(50) 


Useful  Work 


=  Q(ffi  +  h){ i  -  ~  tan  ^  tan  y \  (51) 
Efficiency 

=I*2i<&)=I-|tan^an7'-(52) 

Amount  Q  of   Water  passing 
through   Turbine 

=  inRdiVr"  =  2itRd<i  Vi  sin  ft 
=  2itRd<iUi  sin  ft 

—  2TtR 


n/?.  (53) 

Pressure-head  at  Inlet-surface 


2g 


sin 


(54) 


by  equations  44  and  47. 

When  the  turbine  is  work- 
ing freely  in  space  there  will 
be  no  inflow  of  air  if  pl  >/„, 
i.e.,  if 

Hi          d±   sin  ft 

H\  -\-k      d\  sin  2y' 

When  the  turbine  is 
drowned,  with  a  head  h'  of 
water  over  the  outlet-surface, 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS. 


there  will  be  no  back-flow  of 
water  if 


*  ^  ^  _u  h' 
—  >  —  r  "•  » 

IV          IV     ' 


that  is,  if 


Ti  —  W_       r-^d-j  sin  ft 
r^di  Sin  2*y' 


HI 


there  will  be  no  back-flow  of 
water  if 


A  .   A  ,    ,, 
—  >  --  h  h  , 

' 


that  is,  if 


H,  -  h1      d*  sin  ft 
Hi  -j-  A'      di  sin  2y 


IN  IMPULSE  TURBINES. 


In  an  I.  F.  or  O.  F.  : 

Speed  of  Turbine. 


Since 


t      .     .     (55) 


Velocity  of  Efflux. 


jn 

•U-?  =  4«22  sin2-- 

2. 


*-$#       ft-  . 

cos2  — 

2 


Efficiency 


.    (59) 


In  an  A.  F.  : 

Speed  of  Turbine. 
Since 


(55) 


^  ^.    .     .     (56) 
Velocity  of  Efflux. 


.    (57) 


Useful  Work 


-^  cog2r  Ms8) 

(  3S  2    ) 

Efficiency 

H\      d^  sin2  v 
=  I  ~   LJ-    i    /.  T^ *'     (59) 


298  H  YDRA  UL ICS. 

The  great  advantages  possessed  by  turbines  over  vertical 
wheels  on  horizontal  axes  are  shown  by  a  consideration  of  the 
expressions  for  the  useful  work  and  efficiency.  The  former 
involves  the  available  head  only,  while  the  latter  is  independent 
even  of  that.  Thus  a  turbine  will  work  equally  well  under 
water  or  above  water,  while  its  efficiency  remains  the  same, 
whatever  the  available  head  may  be. 

The  efficiency,   also,   increases  as  the  ratio  — •  diminishes. 

a, 

The  value  of  dl ,  however,  must  not  be  too  small,  as  there  might 
be  a  loss  of  energy  due  to  a  contracted  section  at  entrance, 
while  if  dz  is  made  too  large,  the  vane-passages  will  no  longer 
run  full  bore. 

Finally,  the  efficiency  -increases  as  the  angles  /?  and  y 
diminish. 

In  practice  y  usually  ranges  from  10°  to  30°  in  an  I.  F., 
and  from  20°  to  50°  in  an  O.  F.  and  A.  F.,  an  average  value  being 
20°  for  an  I.  F.,  and  25°  for  an  O.  F.  and  A.  F. 

In  an  I.  F.  ft  generally  ranges  from  135°  to  150°  if  ?/2  —  F2 , 
or  from  30°  to  45°  if  vj'  —  o,  and  in  an  O.  F.  and  P.  F.  from  20° 
to  30°,  an  average  value  being  145°  or  35°  for  an  I.  F.,  accord- 
ing as  #2  =  F2 ,  or  vjr  =  o,  and  25°  for  an  O.  F.  and  A.  F. 


25.  Remarks  on  the  Centrifugal  Head 

From  equations  14  and  17 


In  an  I.  F.  wa  <  u, ,  and  the  term  — L  is  negative. 

Hence  the  velocity  vl  diminishes  as  the  speed  of  the  tur- 
bine increases  and  vice  versa.  The  centrifugal  head  -J— - 

therefore  tends  to  secure  a  steady  motion  in  the  case  of  an  I.  F., 
and  also  to  diminish  the  frictional  loss  of  head.  For  this  rea- 
son it  should  be  made  as  large  as  possible  consistent  with 

practical  requirements,  and  —  is  usually  made  equal  to  2. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.   299 

In  an  O.  F.,  on  the  other  hand,  u^  >  ul  and  the  centrifugal 
head  is  positive.  The  velocity  vl  will,  therefore  increase  and 
diminish  with  the  speed  of  the  turbine  («).  Thus  the  cen- 
trifugal head  is  adverse  to  a  steady  motion,  and  tends  both  to 

augment  a  variation   from   the  normal  speed   and  to  increase 

u  *  _  u  a 
the  frictional  loss  of  head.     It  follows  that  —  —   —  -  should  be 

tg 
as  small  as  possible  consistent  with  practical  requirements,  and 

a  common  value  of  —  is  1.25. 
^i 

Again,  eq.  5  shows  that  the  velocity  of  flow  vr  (and  there- 
fore also  £»,)  increases  as  the  size  of  the  wheel  diminishes,  and 
is  accompanied  by  a  corresponding  increase  in  the  frictional  loss 
of  head.  Hence  it  would  seem  advisable  to  employ  large 
wheels  ;  but  if  the  size  of  a  wheel  is  increased,  it  must  be 
borne  in  mind  that  the  skin-friction  (if  the  turbine  works  under 
water),  the  weight,  and  consequently  the  journal  friction,  will 
all  increase.  Belanger  has  suggested  that  the  efficiency  of  an 
A.  F.  may  be  increased  by  so  forming  the  vane-passages  that 
the  path  of  a  fluid  particle  gradually  approaches  the  axis  of 
rotation. 

26.  Practical  Values  of  the  Velocities,  etc.—  Let  v  be 
the  theoretical  velocity  due  to  the  head  H\  i.e.,  let  v*  =  2gH. 

Experience  indicates  that  the  following  values  will  give 
good  results  in  reaction  turbines  : 


Inl.R,   Vr'  =  Vr"  =      ; 


In  O.  F.,  vr'  =  -  ;  vr"  =  .2iv  to  .172;  ;    u,  =  -u^  =  .$6v. 
4  ri 


In  A.  F.,  vrr  =  vr"  =  .i$v  to  .2v  ;  u,  =  u9  =  -v  to  -v. 

Again,  in  reaction  and  impulse  turbines  the  thickness  of. 
the  vanes  varies  from  -J  inch  to  f  inch  if  of  wrought  iron,  and 


3OO  HYDRAULICS. 

from  \  inch  to  f  inch  if  of  cast  iron.     In  the  latter  case  the 
vanes  are  usually  tapered  at  the  ends. 

In   axial-flow  turbines  the  mean  radius  R  is  often  made  to 
vary 

o      .  _  .  _ 

from  -  yAJ  sin  y  to  2  yAt  sin  y  if  A^  sin  y  <  2  square  feet  ; 


from  --'\fA1  sin  y  to  -\A4,sin  y\i  A1s\ny  >  2sq.  ft.<  l6sq.  ft.; 
4  2 

from  \/  '  Al  sin  ;/  to  —^\/A1  sin  ^  if  ^4,  sin  y  >  16  square  feet. 
4 

In  axial-impulse  turbines  the  mean  radius  R  is  often  made 

to  vary  from  --v/^sin  ;/  to  2<\fA1s'my. 
4 

Also,  the  depth  h  of  the  wheel  varies  from  -  r  to  -  -  but 

o          II 

must  be  determined  by  experience. 
Again, 


For  a  delivery  of  30  to  60  cubic  feet  and  a  fall  of  25  ft.  to 
40  ft.  y  should  be  15°  to  18°,  and  (3  should  be  13°  to  16°. 

For  a  delivery  of  40  to  200  cubic  feet,  and  a  fall  of  5  ft.  to 
30  ft.  y  should  be  1  8°  to  24°,  and  fi  should  be  16°  to  24°. 

For  a  delivery  of  more  than  200  cubic  feet,  and  lower  falls, 
y  should  be  24°  to  30°,  and  0  24°  to  28°. 

In  axial-impulse  turbines  it  may  also  be  assumed  as  a  first 
approximation  that 

.  ?A      vju. 

work  per  pound  =  —  -  =  _^L_J 

2T         g 
and  therefore 

Vl  =  2#,  cos  y  =  2  Vi  cos  y. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    30 1 

27.  Theory  of  the  Suction  (or  Draught)  Tube. — Vortex 
and  axial-flow  turbines  sometimes  have  their  outlet  orifices 
opening  into  a  suction  (or  draft)  tube  which  extends  down- 
wards and  discharges  below  the  surface  of  the  tail-water.  By 
such  an  arrangement  the  turbine  can  be  placed  at  any  conven- 
ient height  above  the  tail-water  and  thus  becomes  easily  acces- 
sible, while  at  the  same  time  a  shorter  length  of  shafting  will 
suffice.  The  suction  tube  is  usually  cylindrical  and  of  constant 
diameter,  so  that  there  is  an  abrupt  change  of  section  at  the 
outlet  surface  of  the  turbine,  producing  a  corresponding  loss  of 
energy  by  eddies,  etc.  This  loss  may  be  prevented  by  so  form- 
ing the  tube  at  the  upper  end  that  there  is  no  abrupt  change 
of  section,  and  by  gradually  increasing  the  diameter  downwards. 
The  cost  of  construction  is  greater,  but  the  action  of  the  tube 
is  much  improved. 

Let  h'  be  the  head  above  the  inlet  orifices  of  the  wheel. 

Let  h"  be  the  head  between  the  inlet  orifices  and  the  sur- 
face of  the  tail-water. 

Let  Ll  be  the  loss  of  head  up  to  the  inlet  surface. 

Let  L^  be  the  loss  of  head  between  the  wheel  and  the  tube 
outlet. 

Let  v^  be  the  velocity  of  discharge  from  the  outlet  at 
bottom  of  tube. 

Let  P  be  the  atmospheric  pressure. 

Then,  assuming  that  there  is  no  sudden  change  of  section 
at  the  outlet  surface, 

h'  ~~=  L' 


and  therefore 

w  2g 

v* 
-       —  2gi  —  K  +  J** 


302  HYDRA  ULICS. 

where  H  =  h'  +  h"  =  total  head  above  tail-water  surface  ;  and 
-^aa,_^42,  Z-j-,  Za  are  expressed  in  the  forms 


2   l  '      4   1  '      *2g'      *2g* 

*3>  /*4>  A*6»  A<6  being  empirical  coefficients. 
Again,  the  effective  head 


and  is  entirely  independent  of  the  position  of  the  turbine  in 
the  tube. 

Also,  if  A  i  is  the  area  of  the  outlet  from  the  suction-tube, 

A^VI  =  Q  =  Alvl  sin  y, 

so  that  v.  can  be  expressed  in  terms  of  z/4,  and  hence  **1  ~  ^  is 

w 

also  independent  of  the  position  of  the  turbine  in  the  tube. 

Suppose  the  velocity  of  flow  to  be  so  small  that  ^4,  v»  L9 
may  be  each  taken  equal  to  nil.     Then 


W 

and  since  the  minimum  value  of  /,  is  also  nil,  the  maximum 
theoretical  height  of  the  wheel  above  the  tail-water  surface  is 
equal  to  the  head  due  to  one  atmosphere.  Again, 


V  3 

=  vl  cos  yul  —  u^u,  —  F,  cos  ft)  +  —  L- 
But 

Alvl  sin  y  =  Q  =  A^  sin  d  =  A^  sin  ft  =  Apt  ; 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    303 

and  hence,  taking 


gH  =  z/,(«i  cos  Y  +  ****•  u*  cos  ft)  —  «•"  +  ^-^ 
and  therefore 

-  w  a  _j_  ?7,  «i  cos  r  +  ^  -  «,  cos  /? 


=  v?  +  2v ^  .  —  cos  y  +  i/^,  .  cos 


—  —  (—  cos  ^  +  V/*8 

(      \  ^ 


where  B  —  —  —  cos          V*8  cos 


Hence  it  follows  that  z/,  increases  with  «a,  i.e.,  with  the 
speed  of  the  turbine,  if 


A  suction-tube  is  not  used  with  an  outward-flow  turbine, 
but  a  similar  result  is  obtained  by  adding  a  surrounding  sta- 
tionary casing  with  bell-mouth  outlet.  A  similar  diffusor  might 
be  added  with  effect  to  a  Jonval  working  without  a  suction-tube 
below  the  tail-water.  The  theory  of  the  diffusor  is  similar  to 
that  of  the  suction-tube. 

28.  Losses  and  Mechanical  Effect. — The  losses  may  be 
enumerated  as  follows: 

I.  The  loss  (Z,)  of  head  in  the  channel  by  which  the  water 
is  taken  to  the  turbine. 

L   -/-^ 
*'  "7l  m  2g> 

fi  being  the  coefficient    of  friction  with  an  average  value  of 


304  HYDRA  ULICS. 

.0067,  /  the  length  of  the  channel  of  approach  tn  its  mean 
hydraulic  depth,  and  v0  the  mean  velocity  in  the  channel. 

Ll  is  generally  inappreciable  in  the  case  of  turbines  of  the 
inward-  and  axial-flow  types,  as  they  are  usually  supplied  with 
water  from  a  large  reservoir  in  which  VQ  is  sensibly  nil. 

If  AQ  is  the  sectional  area  of  the  supply-channel,  then 

A0v0  =  Q  =  A1v1  sin  yy 
and 

£,  =  /,  - 


A, 

II.  The  loss  (Za)  of  head  in  the  guide-passages. 
This  loss  is  made  up  of : 

(a)  The  loss  due  to  resistance  at   the  entrance  into  the 
passages ; 

(b)  The  loss  due  to  the  friction  between  the  fluid  and  the 
fixed  blades; 

(c)  The  loss  due  to  the  curvature  of  the  blades  ; 

(d)  The  loss  of  head  on  leaving  the  guide-passages. 
These  four  losses  may  be  included  in  the  expression 

/a  being  a  coefficient  which  has  been  found  to  vary  from  .025 
to  .2  and  upwards.  An  average  value  of  f9  is  .125,  but  this  is 
somewhat  high  for  good  turbines. 

Note. — In  Impulse  turbines  /a  has  been  found  to  vary  from 
.11  to  .17. 

III.  The  loss  (Z,3)  due  to  shock  at  entrance  into  the  wheel. 
In  order  that  there  may  be  no  shock  at  entrance,  the  relative 
velocity  (  F,)  must  be  tangential  to  the  lip  of  the  vane.     For 

any  other  velocity  (z//  =  ac'}  and  direc- 
tion (dad  =  yf)  of  the  water  at  en- 
trance, evidently 

L3  =  the  loss  of  head 


FIG.  191. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    3O$ 
(v'  sin  y'  —  ^  sin  y)3       (v'  cos  y'  —  vl  cos  y)9 

_  (vf  sin  ;/  —  Vl  sin  <*)a       (z/  cos  y'  —  z\  —  Vl  cos  a)8 

Generally  a?  is  small,  and  L3  is  always  nil  when  the  turbine 
is  working  at  full  pressure  and  at  the  normal  speed. 

This  loss  of  head  in  shock  caused  by  abrupt  changes  of  sec- 
tion, and  also  at  an  angle,  may  be  avoided  by  causing  the  sec- 
tion to  vary  gradually,  and  by  substituting  a  continuous  curve 
for  the  angle. 

IV.  The  loss  (Z,4)  of  head  due  to  friction,  etc.,  in  passing 
through  the  wheel-passages,  including  the  loss  due  to  leakage 
in  the  space  between  the  guides  and  the  inlet-surface.  This 
loss  is  expressed  in  the  form 


V: 


sn 


ftl 


where  f^  varies  from  .10  to  .20. 

Note. — The  loss  of  head  due  to  skin-friction  often  governs 
the  dimensions  of  a  turbine,  and  renders  it  advisable,  in  the  case 
of  high  falls,  to  employ  small  high-speed  turbines. 

V.  The  loss  of  head  (Lb)  due  to  the  abrupt  change  of  sec- 
tion between  the  outlet-surface  and  the  suction-tube. 

As  in  III,  v9  (=ffy  is  suddenly  changed  into  vt'  (•=  fh'\ 
and  loss  of  head  is 


2g  2g  2g 

since  h  '  x  is  very  small  and  may  be  disre- 
garded.    Thus, 


(FiG.  192. 


4  = 


#/  being  the  component  of  vj  (fhf)  in  the  direction  of  the 

axis  of  the  suction-tube. 


3O6  HYDRA  ULICS. 

If  there  is  no  abrupt  change  of  section  between  the  outlet- 
surface  and  the  tube,  Z&  is  nil. 

VI.  The  loss  of  head  (L6)  due  to  friction  the  in  suction-tube. 
Assume  that  the  velocity  v^  of  flow  in  the  tube  is  equal  to  v^ 
the  velocity  with  which  the  water  leaves  the  turbine.  Also  let 
A  be  the  sectional  area  of  the  tube.  Then 


/        f-    f 
6  ~~/6  m'  2g  ~  /6  m'  \      A,      I  2g  ' 

/6(  =/t)  being  the  coefficient  of  friction  with  an  average  value 
of  .0067,  I'  the  length  of  the  tube,  and  m'  its  mean  hydraulic 
depth. 

VII.  The  loss  (Z7)  of  head  due  to  entrance  to  sluice  at  base 
of  tube.     This  loss  may  be  expressed  in  the  form 


A 


the  average  value  of/7  being  about  .03. 

VIII.  The  loss  (Z8)  of  head  due  to  the  energy  carried  away 
by  the  water  on  leaving  the  suction-tube. 


and  z>4  usually  varies  from  |  V2gH  to  f  V2gH. 

In  good  turbines  the  loss  should  not  exceed  6#.  It  might 
be  reduced  to  3$,  or  even  to  i$,  but  this  would  largely  increase 
the  skin-friction. 

IX.  The  loss  of  head  (L9)  produced  by  the  friction  of  the 
bearings. 


being  the  coefficient  of  journal  friction,  Wthe  weight  of  the 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS. 


turbine  and  of  the  water  it  contains,  and  p  the  radius  of  the 
journal. 

Hence  the  total  loss  of  head 


and  the  total  mechanical  effect 


Note.  —  If  there  is  no  suction-tube,  £6  =  O  =  L6  =  L,  = 
and  the  total  loss  becomes 


fall    from   outlet-surface 
tail-water  surface. 


to 


29.  Centrifugal  Pumps. — If  an  hydraulic  motor  is  driven 
in  the  reverse  direction,  and  supplied  with  water  at  the  point 
from  which  the  water  originally  proceeded,  the  motor  becomes 
a  pump.  All  turbines  are  reversible,  and  may,  therefore,  be 
converted  into  pumps,  but  no  pump  has  yet  been  constructed 
of  an  inward-flow  type.  The  ordinary  centrifugal  pump,  Fig. 
193,  is  an  outward-flow  machine. 
It  is  more  economical  and  less 
costly  for  low  falls  than  a  recip- 
rocating pump,  and  has  been 
known  to  give  good  and  eco- 
nomic results  for  falls  as  great 
as  40  feet. 

With  compound  centrifugal 
pumps  very  much  greater  lifts 
are  economically  possible. 

There  are  three  main  differ- 
ences between  centrifugal  pumps 
and  turbines: 

ist.   The  gross  lift   with   a   pump  is  greater,  on  account 


FIG.  193. 


308 


HYDRAULICS. 


of  frictional  resistances,  than  the  fall  in  the  case  of  a  tur- 
bine. 

2d.  The  water  enters  the  pump-fan  without  any  velocity 
of  whirl  (vj  —  o)  and  leaves  the  fan  with  a  velocity  of  whirl 
(vw")  which  should  be  reduced  to  a  minimum  in  the  act  of 
lifting,  but  which  is  by  no  means  small.  In  a  turbine,  on  the 
other  hand,  the  water  has  a  considerable  velocity  of  whirl  (vw'} 
at  entrance,  while  at  exit  the  velocity  of  whirl  (vw")  is  reduced 
to  a  minimum,  and  is  generally  nil. 

3d.  In  a  turbine  the  direction  of  the  water  as  it  flows 
into  the  wheel  is  controlled  by  guide-blades ;  whereas  in  the 
case  of  a  pump,  the  direction  of  the  water,  as  it  flows  towards 
the  discharge-pipe,  is  controlled  by  a  single  guide-blade,  which 
forms  the  outer  surface  of  the  volute,  or  chamber,  into  which 
the  water  flows  on  leaving  the  fan. 


FIG.    194. — Experimental   Centrifugal   Pump   in   the    Hydraulic    Laboratory, 

McGill  University. 

Before  the  pump  can  be  put  into  action  it  must  be  filled, 
and  this  can  be  effected  through  an  opening  (closed  by  a  plug) 
in  the  casing  when  the  pump  is  under  water,  or,  if  the  pump 
is  above  water,  by  creating  a  vacuum  in  the  pump-case  by 
means  of  an  air  pump  or  a  steam-jet  pump,  when  the  water 
must  necessarily  rise  in  the  suction-tube. 

At  first  the  water  rotates  as  a  solid  mass,  and  delivery  com- 
mences when  the  speed  is  such  that  the  head  due  to  centrifugal 

force  r»  —u*\  exceeds  the  lift.     This  speed    may  be  after- 
\      2g      I 

wards  reduced,  providing  a  portion  of  the  energy  is  utilized 
at  exit. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    309 

As  soon  as  the  pump,  which  is  keyed  on  to  a  shaft  driven 
by  a  belt  or  by  gearing,  commences  to  work,  the  water  rises 
in  the  suction-tube  and  divides  so  as  to  enter  the  eye  of  the 
pump-disc  on  both  sides.  As  in  turbines,  the  revolving  pump- 
disc  is  provided  with  vanes  curved  so  as  to  receive  the  water 
at  the  inlet-surface,  for  a  given  normal  condition  of  working, 
without  shock.  Experiment  has  also  tended  to  show  that  the 
angle  between  the  tangents  to  a  vane  and  the  disc  circumfer- 
ence at  the  outlet-surface,  may  be  advantageously  made  as 
small  even  as  15°,  but  manufacturers  hold  different  opinions  on 
this  point.  The  water  leaves  the  disc  with  a  more  or  less  con- 
siderable velocity,  and  impinges  upon  the  fluid  mass  flowing 
round  the  volute,  or  spiral  casing  surrounding  the  disc,  towards 
the  discharge-pipe.  This  volute  should  have  a  section  gradu- 
ally increasing  to  the  point  of  discharge,  in  order  that  the 
delivery  across  any  transverse  section  of  the  volute  may  be 
uniform.  This  volute  is  also  so  designed  as  to  compel  rotation 
in  one  direction  only,  with  a  velocity  corresponding  to  the 
velocity  of  whirl  (vwff)  on  leaving  the  fan.  There  are  exam- 
ples of  pumps  in  which  the  delivery  is  effected  in  all  direc- 
tions, and  the  water  is  guided  to  the  outlet  by  a  number  of 
spiral  blades. 

In  these  pumps  an  important  advantage  is  gained  by  the 
addition  of  a  vortex  or  whirlpool  chamber  surrounding  the 
pump-disc.  The  water  discharged  from  the  disc  then  contin- 
ues to  rotate  in  this  chamber,  and  a  portion  of  the  kinetic 
energy  is  thus  converted  into  pressure  energy,  which  would 
otherwise  be  largely  wasted  in  eddies  in  the  volute  or  discharge- 
pipe.  The  water  leaves  the  vortex  chamber  with  a  diminished 
whirling  velocity  which  cannot  be  very  different  in  direction 
and  magnitude  from  the  velocity  of  the  mass  of  water  in  the 
volute.  The  vortex  chamber  is  provided  with  guide-blades 
following  the  direction  of  free  vortex  stream-lines  (equiangular 
spirals)  so  as  to  prevent  irregular  motion.  A  conical  suction- 
pipe  is  advantageous,  as  it  allows  of  a  gradual  increase  of 
velocity,  and  a  still  greater  advantage  is  to  be  found  in  the 
use  of  a  conical  discharge-pipe.  The  velocity  in  the  dis- 
charge-pipe should  not  be  too  great,  as  it  leads  to  a  waste  of 


310  HYDRAULICS. 

energy.     A  velocity  of  3  to  6  feet  is  found  to  give  the  best 
results. 

Pumps  work  under  different  conditions  from  turbines,  and 
hence  there  are  corresponding  differences  necessary  in  their 
design.  They  work  best  for  the  particular  lift  for  which  they 
are  designed,  and  any  variation  from  this  lift  causes  a  rapid 
reduction  in  the  efficiency. 

30.  Theory  of  Centrifugal  Pump.— 
Denote  the  velocities  at  the  inlet-  and  out- 
let-surfaces of  the  pump-fan  by  the  same 
symbols  as  in  turbines. 

Let  Q  be  the  delivery  of  the  pump. 
Let  Hs  be  the  gross  lift,  including  the 
actual  lift  (ffa),  the  head  due  to  the  velocity 
FIG.  195.  Of  delivery,  the  heads  due  to  the  frictional 

resistances  in  the  ascending  main,  in  the  suction-pipe  and   in 
the  wheel-passages,  and  the  head  corresponding  to  the  losses 
"  in  shock  "  at  entrance  and  exit. 
Let  Ha  be  the  actual  lift. 
The  total  work  done  on  the  wheel 


The  useful  work  done  by  the  pump  — 
Hence 

the  efficiency  (rf)  =          g    g 


At  the  inlet-surface  the  flow  is  usually  radial,  so  that  y  =  90°, 
and  the  velocity  of  whirl  vj  is  nil. 
Thus, 


the  efficiency  =      fr*  =  77, 


and  the  equation 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL  PUMPS.    311 

is  the  fundamental  equation  governing  the  design  of  centrifu- 
gal pumps. 
Again, 

H: 

the  efficiency  n  =  —  jh-  =  i 
" 


For  a  given  speeed  («3)  this  is  a  maximum  when 

j          (VJJ+(U,-V    "y 

y       v       y 


-77  =  a  minimum  —  -- 

vw  w  w 

Hence,  differentiating,  - 

u*  tan2  ft 

-  -<^-  +  sec-  />  =  Q, 

and  therefore 

vjf  =  u^  sin  ft, 

and  is  the  velocity  of  whirl  at  exit  which,  for  a  given  speed  (wa), 
will  give  a  maximum  efficiency. 
Note.—li  u^  =  vw",  then 


and  the  water  leaves  the  fan  with  a  velocity  equal  to  that  due 
to  at  least  one  half  of  the  gross  lift.  The  efficiency  must 
therefore  be  necessarily  less  than  .5. 

Again,  since  vr"  cot  ft  =  u^  —  vw",  ft  must  be  90°  if  u^  =  vj'-, 
but  ft  is  generally  much  less  than  90°,  and  therefore  vw"  is 
generally  less  than  uy  Let  vw"  =  ku^>  k  being  an  empirical 
coefficient  less  than  unity. 


Then  kit?  —  gHe  and  the  efficiency  =  -~> 

KU^ 

Consider  two  cases. 

CASE  I.  Pump  without  a  vortex-chamber. 

When  the  water  is  discharged  into  the  volute,  the  velocity 
of  flow  (vr")  is  wasted  and  the  velocity  of  whirl  (vw'f)  is  sud- 
denly changed  to  the  velocity  vs  of  the  mass  of  water  in  the 


312  HYDRAULICS. 

volute  assumed  to  be  moving  in  a  direction  tangential  to  the 
pump-disc.     Thus, 

(yjy  -  far    (vjf  -  vy 

the  gam  of  pressure-head  =  - 


i  fe/')9  ^  " 

which  is  a  maximum  and  equal  to—  when  vs  =  -^—  . 

4     g 

This  gain  of  head  is  always  very  small  and  may  be  dis- 
regarded as  being  almost  inappreciable.  Neglecting  also  the 
losses  due  to  frictional  resistances,  etc.,  then,  precisely  as  in 
the  case  of  turbines, 

v±_    ,    TT  __    f  variation  of  pressure-head  between 
2g  (      outlet  and  inlet  surfaces. 

*.*-«.•    FV-F? 


But  V?  =  u?  +  T^2,  since  y  =  90°,  and  therefore 

_  u*  ~  ~_¥JL.  _  u*  ~  (ui  ~  v™'}*  sec2  ft 

and 

u  2  —  (u  —  v  //)2  sec2  ft 
the  efficiency  —  -  w.. 

2«^w" 

which   is   a   maximum    for   a   given   speed  &a   and    equal   to 
; — j- — : — -5  when  vwff  =  u^  sin  /?. 

Thus  the  efficiency  increases  as  ft  diminishes. 

When  ft  =  90°,  or  ^wr/ —  &2,  the  maximum  efficiency  is  £, 
and  therefore  one  half  of  the  work  done  in  driving  the  pump 
is  wasted. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    313 

Note.  —  Loss  of  head 

=  loss  due  to  hydraulic  friction 

-f-  loss  due  to  abrupt  change  from  vw"  to  v, 

-\-  loss  due  to  dissipation  of  vr" 

-f-  loss  due  to  vs  carried  away 
=  loss  due  to  friction  (hydraulic) 

+  *,(*„"  -  ^        (*l_  _     (VT 

2g  2g  2g 

=  loss  due  to  friction  (hydraulic) 

,  (*.")•  , 
~~ 


when  vs  =  \vj'. 

CASE  II.  Pump  ivith  a  vortex-chamber  (Fig.  199). 

The  diameter  (—  2r3)  of  the  outer  surface  of  this  chamber 
should  be  at  least  twice  that  of  the  outlet-surface  of  the  pump- 
disc. 

Assuming  that  the  motion  in  the 
•chamber  is  a  free  vortex,  then 

the  gain  of     )  _  v^_  I         r?\ 
pressure-head  )        2g  \         r32/ 

and  hence 


the  efficiency  = 


T,,  .  .  FIG.  106. 

This,   again,   is  a  maximum  for 

a  given  speed,  when  vj  =  u^  sin  fiy  its  value  being 

I  +'(l  -  Sj)  sin  ft 

I  +  sin  ft 


3 1 4  HYDRA  ULICS. 

This  expression  increases  as  ft  diminishes,  but  the  value  of 
ft  is  not  of  so  much  importance  as  in  Case  I,  and  it  is  very 
common  to  make  ft  equal  to  30°  or  40°. 

When  ft  =  90°  the  maximum  efficiency  =  -  ( 2  -    -M  =  — 

if  ra  =  2r,. 

31.  Practical  Values. — The  following  values  are  often 
adopted : 


3  =  d^  when  faces  of  pump-disc  are  parallel ; 
^  =  \d^  when  pump-disk  is  coned. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    31$ 


EXAMPLES. 

1.  An  accumulator  ram  is  9  inches  diameter  and  21  feet  stroke.     Find 
the  store  of  energy  in  foot-pounds  when  the  ram  is  at  the  top  of  its 
stroke,  and  is  loaded  till  the  pressure  is  750  Ibs.  per  square  inch. 

Ans.  958,000  ft.-lbs. 

2.  In  a  differential  accumulator  the  diameters  of  the  spindle  are  7 
inches  and  5  inches  ;  the  stroke  is  10  feet.    Find  the  store  of  energy  when 
full  and  loaded  to  2000  Ibs.  per  square  inch.  Ans.  377,000  ft.-lbs. 

3.  A  direct-acting  lift  has  a  ram  9  inches  diameter,  and  works  under 
a  constant  head  of  73  feet,  of  which  13  per  cent,  is  required  by  ram-fric- 
tion and  friction  of  mechanism.     The  supply-pipe  is  100  feet  long  and  4 
inches  diameter.     Find  the  speed  of  steady  motion  when  raising  a  load 
of  1350  Ibs.,  and  also  the  load  it  would  raise  at  double  that  speed. 

If  a  valve  in  the  supply-pipe  is  partially  closed  so  as  to  increase  the 
coefficient  of  resistance  by  5!,  what  would  the  speed  be  ? 

Ans.  Speed  =  2  ft.  per  second  ;  load  =  150  Ibs. 

4.  Eight  cwt.  of  ore  is  to  be  raised  from  a  mine  at  the  rate  of  900  feet 
per  minute  by  a  water-pressure  engine,    which  has  four  single-acting 
cylinders,  6  inches  diameter,  18  inches  stroke,  making  60  revolutions  per 
minute.     Find  the  diameter  of  a  supply-pipe  230  feet  long  for  a  head 
of  230  feet,  not  including  friction  of  mechanism. 

Ans.  Diameter  =•  4  inches. 

5.  If  A.  be  the  length  equivalent  to  the  inertia  of  a  water-pressure 
engine,  F  the  coefficient  of  hydraulic  resistance,  both  reduced  to  the 
ram,   -z/o  the  speed   of   steady  motion,   find   the   velocity  of   ram  after 
moving  from  rest  through  a  space  x  against  a  constant  useful  resistance. 
Also  find  the  time  occupied. 

Ans.  v*  = 

—V 

6.  An  hydraulic  motor  is  driven  from  an  accumulator,  the  pressure 
in  which  is  750  Ibs.  per  square  inch,  by  means  of  a  supply-pipe  900  feet 
long,  4  inches  diameter;  what  would  be  the  maximum  power  theoreti- 
cally attainable,  and  what  would  be  the  velocity  in  the  pipe  correspond- 
ing to  that  power?    Find  approximately  the  efficiency  of  transmission  at 
half  power.  Ans.   H.P.  =  240 ;  v  =  22  ft. ;  efficiency  =  .96  nearly. 

7.  A  gun  recoils  with  a  maximum  velocity  of  10  feet  per  second. 
The  area  of  the  orifices  in  the  compressor,  after  allowing  for  contraction, 
may  be  taken  as  one  twentieth  the  area  of  the  piston.     Find  the  initial 
pressure  in  the  compressor  in  feet  of  liquid. 


HYDRAULICS. 

Assuming  the  weight  of  the  gun  to  be  12  tons,  friction  of  sUde  3 
tons,  diameter  of  compressor  6  inches,  fluid  in  compressor,  water,  find 
the  recoil. 

Find  the  mean  resistance  to  recoil.  Compare  the  maximum  and 
mean  resistances,  each  exclusive  of  friction  of  slide. 

Ans.  621;   4ft.   2^  in. ;    total  mean  resistance  =  4.4  tons; 
ratio  =  2.5. 

8.  A  reaction  wheel  is  inverted  and  worked  as  a  pump.     Find  the 
speed  of  maximum  efficiency  and  the  maximum  efficiency,  the  coeffi- 
cient of  hydraulic  resistance  referred  to  the  orifices  being  .125. 

Ans.  Speed  =  twice  that  due  to  lift ;  .758. 

9.  A  reaction  wheel  with  orifices  2  in.  in  diameter  makes  80  revolu- 
tions per  minute  under  a  head  of  5  ft.     The  distance  between  the  centre 
of  an  orifice  and  the  axis  of  rotation  is  12  inches.     Find  the  H.P.  and 
the  efficiency.  Ans.  .146;  .596. 

10.  In  a  reaction  wheel  the  speed  of  maximum  efficiency  is  that  due 
to  the  head.     In  what  ratio  must  the  resistance  be  diminished  to  work 
at  |  this  speed,  and  what  will  then  be  the  efficiency?     Obtain  similar 
results  when  the   speed   is   diminished   to   three   fourths   its   original 
amount.  Ans.  .949;  .8896;  1.071;  .753. 

11.  In  a  reaction  wheel,  determine  the  per  cent  of  available  effect 
lost,  (i)  if  i?  =  2gH\  (2)  if  tt*  =  ^gH;  (3)  if  u1  =  ZgH. 

What  conclusion  may  be  drawn  from  the  results? 
Efficiencies  are  respectively  .828,  .9,  .945. 

12.  An  undershot  water-wheel  with  straight  floats  works  in  a  straight 
rectangular  channel  of  the  same  width  as  the  wheel,   viz.,  4  ft.;  the 
stream  delivers  28  cub.  ft.  of  water  per  second,  and  the  efficiency  is  £. 
Find  the  relation  between  the  up-stream  and  down-stream  velocities. 
If  the  velocity  of  the  inflowing  water  is  2  ft.  per  second,  find  the  velocity 
on  the  down-stream  side  and  determine  the  mechanical  effect  of  the 
wheel,  its  diameter  being  20  ft.,  the  diameter  of  the  gudgeons  being  4 
in.,  and  the  coefficient  of  friction  .008. 

13.  A  vane  rotates  about  an  axis  with  an  angular  velocity  A,  and 
and  water  moves  freely  along  the  vane.    Show  that  the  work  per  unit  of 
weight  of  water,  due  to  centrifugal  force,  in  moving  from  a  point  distant 

A1(<y    2  *•   2\ 

r\  ft.  from  the  axis  to  a  point  distant  r-i  ft.  from  the  axis  is  — '— . 

14.  Determine  the  effect  of  a  low  breast  or  undershot  wheel  15  ft.  in 
diameter  and  making  8  revols.  per  minute;   the  fall  is  4  ft.  and  the 
delivery  20 cub.  ft.  per  second;  the  velocity  of  the  stream  before  com- 
ing on  the  wheel  is  double  that  of  the  wheel.  Ans.  1490  ft.-lbs. 

15.  The  efficiency  of  an  undershot  water-wheel  working  in  a  straight 
rectangular  channel  with  horizontal  bed  is  \.    Find  the  relation  between 
the  up-  and  down-stream  velocities  of  flow. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    31  / 

16.  Determine  the  mechanical  effect  of  an  undershot  wheel  of  12  ft. 
diam.  making  10  revols.  per  minute,  the  fall  being  3  ft.  and  the  quantity 
of  water  passed  per  second  15  cub.  ft. 

17.  Ascertain  the  general  proportions    of   a  Poncelet  wheel,  being 
given:    height  of  fall  =  4^   ft.;   delivery  of   water  =  40  cub.    ft.    per 
second;    radius  of   exterior  circumference  =  9  ft.  ;    thickness  of  the 
stream  =  9  in. 

18.  Design  a  Poncelet  wheel  for  a  fall  of  4.5  ft.  and  24  cub.  ft.  of 
water  per  second,  using  the  formulae  on  pages  237-239,  taking  y  =  20°, 
and  also  A  =  20°  as  a  first  approximation. 

Ans.  a  =  143°  57' ;  depth  of  crown  =  1.73  ft. ;  depth  of  stream 
=  .386  ft.  ;  b  =  4.14  ft.;  radius  of  bucket  =  2.35  ft. ;  ^  = 
128°  6' ;  A  =  18.69°;  number  of  buckets  =  44;  mechanical 
effect  =  8.5  H.P. ;  efficiency  =  .69. 

19.  15  cub.  ft.  of  water  per  second  with  a  fall  of  8g  ft.  are  brought  on 
a  breast-wheel  revolving  with  a  linear  velocity  of  5  ft.;  depth  of  shroud- 
ing =  12  in.;  the  buckets  are  half  filled,  and  vi  =  2tt ;  also  r\  —  12  ft. 
Find  the  theoretical  mechanical  effect.  Ans.  7240  ft.-lbs. 

20.  A   wheel  is   to   be  constructed    for  a  3O-ft.  fall  having  an   8-ft. 
velocity  at  circumference  and  taking  on  the  water  at  12°  from  the  sum- 
mit with  a  velocity  of  16  ft.     Determine  the  radius  of  the  wheel  and  the 
number  of  revols.  Ans.   12. 8ft.;  5.94. 

21.  If  for  the  wheel  in  example  20  the  number  of  revols.  is  5,  and 
Vi  =  2u,  the  water  being  again  taken  on  at  12°,  find  the  radius  and  u. 

Ans.  13.98  ft. ;  7.3  ft.  per  sec. 

22.  A  breast  wheel  passes  12  cub.  ft.  of  water  per  second*  and  for  the 
speed  =  3-z/i  =  4  ft.  per  second  the  loss  of  mechanical  effect  due  to  the 
relative  velocity  V  being  destroyed  is  a  minimum.     Find  this  effect. 

23.  In  a  breast-wheel  Q  =  10  cub.  ft.  per  second ;  H  =  10  ft. ;  v\  = 
-£#  ;  u  =  4i  ft.  per  second  ;  y  =  30° ;  diam.  of  gudgeon  =  6  in. ;  diam.  of 
wheel  =  30  ft. ;  /*  =;  .08  ;  weight  of  wheel  and  water  =  20,000  Ibs.     Find 
the  mechanical  effect  of  the  wheel.    (Neglect  loss  of  effect  due  to  escape 
of  water  from  buckets  and  to  frictional  resistance  along  the  curb.) 

24.  The  quantity  of  water  laid  on  a  breast-wheel  by  an  overfall  sluice 
=  6  cub.  ft.  per  second,  the  total  fall  being  4  ft.  6  in.,  and  the  velocity 
of  the  periphery  5  ft.  per  second  ;  also  52/1  =  Su,  and  if  d  be  the  depth  of 
the  shrouding  2bdu  —  $Q  (in  the  present  case  d  =  12  in.).    Find  the  effec- 
tive fall,  the  height  of  the  lip  of  the  guide,  the  angle  of  inclination  at 
the  end  of  the  guide-curve,  the  breadth  of  the  lip  of  the  guide-curve,  and 
the  radius  of  the  wheel   that  the  water  may  enter  tangentially.     If  the 
radius  is  limited  to  12  ft.  6  in.,  find  the  deviation  of  the  direction  of 
motion  of  the  water  from  that  of  the  wheel  at  the  point  of  entrance. 

Ans.  6.9  ft. ;  .325  ft. ;  34°  46' ;  2f  ft.  ;  38.6  ft. ;  28°  36'. 

25.  In   an   overshot  wheel  r\  =  15  ft.,  d  —  10  in.,  ft  =  f^.     If  the 


3l8  HYDRAULICS. 

division  circle  is  at  one  half  of  the  depth  of  the  crown,  find  the  angle 
(yi)  between  the  bucket-lip  and  the  wheel's  periphery. 

Ans.  yi  =  18°  i'. 

26.  An  overshot  wheel  in  which  r\  —  18  ft.  makes  4  revolutions  per 
minute,  and  the  velocity  of  the  water  on  entering  the  buckets  is  twice 
that  of  the  wheel's  periphery.     If  yi  =  20°,  find  a,  and  also  find  the  rela- 
tive velocity  (  V)  of  the  entering  water. 

Ans.  a.  =  10°  9' ;   V  =  7.78  ft.  per  second. 

27.  If  one  fourth  of  the  theoretic  capacity  of  a  bucket  is  filled  by  the 
water,  find  the  greatest  number  of  buckets  theoretically  possible,  the 
depth  of  the  crown  being  i  ft.,  the  radius  (ri)  to  the  outer  periphery  12 
ft.,  the  angle  yi  20°,  and  the  velocity  of  the  entering  water  twice  that  of 
the  wheel's  periphery. 

Ans.     103.1.     Making  allowance  for  exit  of  air,  the  number  of 
buckets  might  be  about  two  thirds  of  this  amount,  or,  say,  69. 

28.  A  wheel  of  3o-ft.  diam.  with  72  buckets  makes  7  revolutions  per 
minute,  Q  being  5  cub.  ft.  per  second.     The  division  circle  is  halfway 
between  the  outer  and  inner  peripheries.     If  d  =  i  ft.  and  vi  =  2u,  find 
the  effect  due  to  impact. 

29.  A  3O-ft.  wheel  weighs  24,000  Ibs.  and  makes  6  revolutions  per 
minute;  its  gudgeons  are  6  in.  in  diameter  and  the  coefficient  of  friction 
is  .08.     The  water  enters  the  wheel  with  a  velocity  of  1 5  ft.  per  second, 
and  in  a  direction  making  an  angle  of  10°  with  the  direction  of  motion 
of  the  wheel  at  the  point  of  entrance.     The  deviation  from  the  summit 
of  the  point  of  entrance  is  12°,  of  the  point  where  spilling  begins  is  150°, 
of  the  point  where  all  is  spilt  is  160°,  and  5  cub.  ft.  of  water  enter  the 
wheel  per  second,  of  which  the  partially  filled  buckets  contain  one  half. 
Determine  the  total  mechanical  effect.  Ans.  9305.6  ft.-lbs. 

30.  The  velocity  of  the  pitch  circle  is  9!  ft.;  the  angle  between  the 
directions  of  motion  of  stream  and  wheel  is  15°.     Find  impulsive  action 
of  wheel.  Ans.  91  ft.-lbs.  per  cub.  ft.  of  water. 

31.  An  overshot  wheel  40  ft.  in  diameter  makes  4  revolutions  per 
minute  and  passes  300  cub.  ft.  of  water  per  minute.     Show  how  to  deter- 
mine the  mechanical  effect  of  the  wheel.     (Neglect  friction  of  gudgeons.) 

If  the  gudgeons  are  6  in.  in  diameter  and  the  wheel  weighs  30,000  Ibs., 
by  how  much  will  the  mechanical  effect  be  diminished  (/=  .008)  ? 

Ans.  25  ft.-lbs.  per  sec. 

32.  The  diameter  of  an  overshot  wheel  =  30  ft. ;  Vi  =  15  ft. ;  u  =  9$ 
ft.,  deviation  of  impinging  water   from  direction   of   motion  of  wheel 
(y)  =  8^° ;  deviation  of  point  of  entrance  from  summit  =  12°  ;  deviation 
of  point  where  spilling  begins  from  the  centre  =  58^° ;  deviation  of  point 
where  spilling  ends  =  70^° ;  Q  =  5  cub.  ft.     Find  total  effect  of  impact 
and  weight.  Ans.  17  H.  P. 

33.  An  overshot  wheel  with  a  radius  of  15   ft.  and  a  i2-in.  crown 
takes    10  cub.   ft.   of  water  per  second  and  makes  5  revolutions  per 


H  YDRA  ULIC  MO  TORS  A  ND    CEN  TRIFUGA  L   P  UMPS.    3 1 9 

minute.     If  m  =  i,  find  the  width  of  the  wheel  and  the  number  of  the 
buckets.  Ans.  5TV  ft. ;  75  or  90. 

34.  An  overshot  wheel  of  32  ft.  diameter  makes  5  revolutions  per 
minute.     Find  the  angle  between  the  water-surface  in  a  bucket  and  the 
horizontal  when  the  lip  is  140°  from  the  summit.  Ans.  4°  33'. 

35.  An  overshot  wheel  of  10  ft.  diameter  makes  20  revolutions  per 
minute.     Find  the  angle  between  the  water-surface  and  the  horizontal 
when  the  lip  is  (i)  90°  from  the  summit,  (2)  45°  26'  from  the  summit. 

Ans.  (i)  34°  16';  (2)45°  26'. 

36.  The  water  enters  an  overshot  wheel  at  12°  from  the  summit  with 
a  velocity  of  16  ft.  per  second  and  the  linear  velocity  of  the  wheel's  pe- 
riphery is  8  ft.  per  second.     The  fall  is  30  ft.     Find  the  diameter  of  the 
wheel  and  the  number  of  revolutions  per  minute. 

Ans.  25.68  ft. ;  5.94. 

37.  An  overshot  wheel  of  36  ft.  diameter  and  with  96  buckets  has  a 
peripheral  velocity  of  7^  ft.  per  second.     The  water  enters  with  a  velocity 
of  15  ft.  per  second  and  acquires  in  the  wheel  a  velocity  of  16.49  ft-  Per 
second   Find  the  distance  through  which  the  float  moves  during  impact. 

Ans.  2.15  ft. 

38.  The  sluice  for  a  lo-ft.  overshot  wheel  is  vertically  above  the  centre 
and  inclined  at  45°  to  the  vertical.     The  water  enters  the  buckets  at  a 
point  2  ft.  vertically  below  the  sluice  and  10°  from  the  summit  of  the 
wheel.     Find  the  angle  between  the  directions  of  motion  of  the  entering 
water  and  of  the  wheel's  circumference.     Also  find  the  velocity  of  the 
water  as  it  enters  the  wheel. 

39.  In  an  overshot  wheel  z>i  =  17  ft.  ;  u  =  n  ft.  per  second;  elbow- 
angle  =  70°;  division-angle  =  5°;  water  enters  the  first  bucket  at  12° 
from  summit  of  wheel.     Find  (a)  the  relative  velocity  Fso  that  water 
may  enter  unimpeded;    (£)   the  direction  of   the  entering  water;    (c) 
the  diameter  of  the  wheel,  which  makes  5  revolutions  per  minute ;  (d) 
the  position  and  direction  of  the  sluice,  which  is  2  ft.,  measured  hori- 
zontally from  the  point  of  entrance. 

40.  In  an  overshot  wheel  the  deviation  of  the  impinging  water  from 
the  direction  of  motion  of  the  wheel  is  10°  ;  the  velocity  (vi)  of  the  im- 
pinging stream  =  15  ft.  per  second;  of  the  circumference  of  the  wheel 
(«)  =  15  cos  10°.     What  proportion  of  the  head  is  sacrificed? 

41.  A  3o-ft.  water-wheel  with  72  buckets  and  a  12-in.  shrouding  makes 
5  revolutions  and  receives  240  cub.  ft.  of  water  per  minute.     Find  the 
width  and  sectional  area  of  a  bucket.     The  fall  is  30  ft. ;  at  what  point 
does  the  water  enter  the  wheel,  the  inflowing  velocity  being  i|  times 
that  of  the  wheel's  periphery?     Also  find  the  deviation  of  the  water- 
surface  from   the  horizontal  at   the   point  at  which  discharging  com- 
mences, i.e.,  140°  from  the  summit. 

42.  What  number  of  buckets  should  be  given  to  an  overshot  wheel  of 


3  2O  H  YDRA  UL ICS. 

40  ft.  diameter  and  12  in.  width  in  wheel,  pitch-angle  =  4°,  thickness  of 
bucket  lip  =  i  in.,  water  area  =  24^  sq.  in.  ? 

43.  A  wheel  makes  5  revolutions  per  minute,  the  radius  is  16  ft.,  and 
the  discharging  angle  50°.     Find  deviation  of  water-surface  from  the 
horizon.  Ans.  4°  .29. 

44.  A  wheel  makes  20  revolutions  per  minute;  radius  =  5  ft.,  angle 
of  discharge  =  o°.     Find  deviation  of  water-surface  from  horizon.    Also 
find  deviation  at  44°  35'  above  centre.  Ans.  4°  33' ;  44°  34'. 

45.  The  water  in  a  head-race  stands  4.66  ft.  above  the  sole  and  leaves 
the  race  under  a  gate  which  is  raised  6  in.  above  the  sole,  the  coefficient 
of  velocity  (v*)  being  .95.    The  water  enters  a  breast  wheel  in  a  direction 
making  an  angle  of  30°  with  the  tangent  to  the  wheel's  periphery  at  the 
point  of  entrance.     The  speed  (u)  of  the  periphery  is  10  ft.  per  second, 
the  breadth  of  the  wheel  is  5  ft.,  the  depth  of  the  water  beneath  the 
axle  is  8  in.,  and  the  length  of  the  flume  is  8.2  ft.     Find  the  loss  of 
head  (a)  due  to  the  destruction  of  the  relative  velocity  (V)  at  entrance; 
(b)   due   to   the   velocity   of  flow  in  the  tail-race ;    (c)    in   the  circular 
flume.  Ans.  (a)  i.u  ft.;  (£)  1.57  ft.  ;  (c]  .44  ft. 

46.  In  the  preceding  example,  find  how  the  losses  of  head  would  be 
modified  if  the  flume  were  lowered  1.03  ft.,  and  if  the  point  of  entrance 
were  raised  so  as  to  make  u  =  v\  cos  30°. 

47.  A  water-wheel  has  an  internal  diameter  of  4  ft.  and  an  external 
diameter  of  8  ft.;  the  direction  of  the  entering  water  makes  an  angle  of 
15°  with  the  tangent  to  the  circumference.     Find  the  angle  subtended 
at  the  centre  of  the  wheel  by  the  bucket,  which  is  in  the  form  of  a  cir- 
cular arc,  and  also  find  the  radius  of  the  bucket. 

48.  An  overshot  wheel  5  ft.  wide,  30  ft.  in  diameter,  having  a  12-in. 
crown  and  72  buckets,  receives   10  cub.   ft.   of  water  per  second  and 
makes  5  revolutions  per  minute.     Determine  the  deviation  from  the 
horizontal  at  which  the  water  begins  to  spill,  and  also  the  corresponding 
depression  of  the  water-surface. 

49.  An  overshot  wheel  makes  —  revolutions  per  minute ;  its  mean 

iTt 

diameter  is  32  ft. ;  the  water  enters  the  buckets  with  a  velocity  of  8  ft. 
per  second  at  a  point  12°  30'  from  the  summit  of  the  wheel.  At  the 
point  of  entrance  the  path  of  the  inflowing  water  makes  an  angle  of  30° 
with  the  horizontal.  Show  that  the  path  is  horizontal  vertically  above 
the  centre.  The  sluice-board  is  placed  at  a  point  whose  horizontal 
distance  from  the  centre  is  one  half  that  of  the  point  of  entrance. 
Find  its  position  relatively  to  the  centre  and  its  inclination  to  the  hori- 
zon. (Sin  12°  30'  =  .2165). 

50.  The  water  enters  the  buckets  of  the  wheel   in  the  preceding 
example  without  shock.     Find  the  elbow-angle.     Also,  if  the  buckets 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    $21 

begin  to  spill  at  150°  from  the  summit,  find  where  the  bucket  is  empty 
and  the  number  of  buckets.  (Depth  of  crown  =  12  in.;  thickness  of 
bucket  =  \\  in.) 

51.  Given  7/1  =  15  ft.  per  second,  and  S  —  2oJ°.     Find  the  position  of 
the  centre  of  the  sluice,  which  is  4  in.  above  the  point  of  entrance. 

Ans.  .097  ft.  vertically  below  and  1.114  ft-  horizontally  from  the 
summit.  The  axis  of  the  sluice  is  inclined  at  9°  58'  to  the 
horizontal. 

52.  In  an  overshot  water-wheel  2/1  =  15  ft.;  u  =  10  ft.  ;  elbow-angle 
=  70^° ;    division-angle  =  4^° ;    deviation  from  summit  of  point  of  en- 
trance =  12°.     Find  the  deviation  of  the  layer  from  that  of  the  arm,  so 
that  the  water  might  enter  unimpeded;  also  find  the  inclination  of  the 
layer  to  the  horizon,  and  the  value  of  V .     If  the  centre  of  the  sluice- 
aperture  is  to   be  4  in.  above  point  of  entrance,  find  its  vertical  and 
horizontal  distance  Trom  the  vertex  of  the  stream's  parabolic  path  which 
is  vertically  above  the  centre  of  the  wheel,  and  also  find  inclination  oi 
sluice-board  to  horizon. 

Ans.  15!° ;  2oJ° ;  5.3  ft.  per  sec. ;  .42  ft. ;  1.04  ft. ;  9°  34', 

53.  In  an  overshot  wheel  Q=  18  cub.  ft.;  r\  =6  ft. ;  d—  i  ft. ;  b  = 
4  ft. ;  N  —  24.     At  the  moment  spilling  commences  the  area  afd  =  1.025 
sq.  ft.;   between  this  point   and  the   point  where   the  spilling  is   com- 
pleted three  buckets  are  interposed,  the  sectional   areas  of  the  water 
being  .591,  .409,  and  .195  sq.  ft.,  respectively.     Find  (a)  the  sectional  area 
of  bucket,  (b)  the  point  where  the  spilling  commences,  (c]  the  point  where 
the  spilling  is  completed,  (d)  the  height  of  the  arc  of  discharge,  (<?)  the 
mechanical  effect  due  to  the  fall  of  the  water  through  the  arc,  of  discharge. 

'  Ans.   (a)  .662  sq.  ft  ;  (b)  0  =  7°  26',  0  =  28°  33' ; 

(c)  e  =  73°  15'.  0  =  5°  59' ;    (d)  449  ft.  I  (')  4-93  H.P. 

54.  In  the  preceding  example,  if  the  water  enters  with  a  velocity  of 
20  ft.  per  second  at  20°  below  the  summit,  and  if  the  direction  of  the 
inflowing  stream  makes  an  angle  of  25°  with  the  wheel's  periphery  at 
the  point  of  entrance,  find  the  mechanical  effect  (a)  due  to  impulse,. 
(b)  due  to  the  fall  to  the  point  where  spilling  commences. 

Ans.  (d)  5.34  H.P.;  (b)  12.15  H.P. 

55.  300  cub.  ft.  of  water  per  minute  enter  the  buckets  of  a  4o-ft. 
overshot  wheel  with  a  12-in.  crown  and  making  four  revolutions  per 
minute.     The  wheel  has   136  buckets.     At  the  moment  when  spilling 
commences  the  area  afd  (Fig.  156)  =  102  sq.  in.,  and  the  area  abed '  = 
24.5  sq.  in.     The  spilling  is  completed  when  the  angle  between  the  hori- 
zontal and  the  radius  to  the  lip  of  the  bucket  =  62°  30'.     Between  these 
two  positions  three  buckets  are  interposed,  the  sectional  areas  of  the 
water  in  the  buckets  being  24.5,  14.48,  and  6.6  sq.  in.,  respectively.     The 
vertical  distance  between  the  water-surface  in  the  first  bucket  and  the 
centre  is  18  ft.     Find  (a)  the  width  of  the  wheel,  (b)  the  cross-section  of 


3 2 2  HYDRA  ULICS. 

a  bucket,  (c)  the  angle  between  the  horizontal  and  the  radius  to  the  lip 
of  the  bucket  when  spilling  commences,  (d)  the  height  of  the  discharg- 
ing arc,  (<?)  the  mechanical  effect  due  to  weight. 

Ans.  (a)  2.4  ft.;    (b)  33.09  sq.   ft.;     (c)  6  =  52°  2*';    (d)  1.9  ft.; 
(<?)  19.48  H.P. 

56.  As  the  bucket  arm   cd  moves  downwards  from  the  horizontal 
position,  show  that  while  the  wheel  moves  through  an  angle  the  last 
particle  of  water  at  c  will  move, through  a  distance  approximately  equal 

?*(/r^*  ~{~  u^ } 
to  -     — ^ (6  —  sin  6),  r  being  the  distance  (assumed  constant)  of  the 

particle  of  water  from  the  axis,  and  u  being  the  linear  velocity  of  the 
wheel  at  the  radius. 

57.  If  the  last  particle  of  water  leaves  the  buckets  just  as  the  lip  d 
reaches  the  lowest  point  of  the  wheel,  and  if  the  arm  is  i  ft.  in  length, 
find  the  angle  between  the  lip"  and  the  wheel's  periphery  (i)  for  a  wheel 
of  20  ft.  diameter,  the  peripheral  velocity  being  5  ft.  per  second  ;  (2)  for 
a  wheel  of  40  ft.  diameter,  the  peripheral  velocity  being  10  ft.  per  second ; 
(3)  for  a  wheel  of  10  ft.  diameter,  the  peripheral  velocity  being  8  ft.  per 
second.  Ans.  (i)  20^° ;  (2)  20° ;  (3)  40°. 

58.  In  an  overshot  wheel  of  30  ft.  diameter,  5  cub.  ft.  of  water  per 
second  enter  the  buckets  with  a  velocity  of  16  ft.  per  second  and  the 
wheel's  velocity  at  the  division  circle  is  7  ft.  per  second.     The  point  of 
entrance  is  18°  from  the  summit,  and  the  angle  between  the  directions 
of  the  inflowing  water  and  the  wheel's  periphery  at  the  point  of  entrance 
is  12°.     The  water  begins  to  spill  at   148^°  from  the  summit  and  the 
spilling  is  complete  at  i6o£°  from  the  summit.     Find  the  total  mechani- 
cal effect  due  to  impulse  and  weight.     What  is  the  tangential  force  at 
the  outer  periphery?  Ans.  16.28  H.P. ;  1194  Ibs. 

59.  20  cub.  ft.  of  water  per  second  enter  an  undershot  wheel  of  30  ft. 
diameter,  making  8  revolutions  per  minute  through  an  underflow  sluice. 
The  velocity  of  the  entering  water  is  twice  that  of  the  wheel's  periphery. 
Find  (a)  the  head  of  water  behind  the  sluice,  (b)  the  fall,  (c}  the  theo- 
retical mechanical  effect,  (d)  the  actual  mechanical  effect,  disregarding 
axle  friction. 

Ans.   (a)  2.779  ft.;  (b)  1.221  ft.;  (c}  5.57  H.P.;  (d)  2.62  H.P. 

60.  20  cub.  ft.  of  water  per  second  enter  a  breast  wheel  of  32  ft.  diam 
eter  and  having  a  peripheral  velocity  of  8  ft.  per  second,  at  an  angle  of 
25^    with  the  circumference.     The  depth  of  the  crown  is  ij  ft.;  the  buc- 
kets are  half  filled,  and  the  fall  is  9  ft.     The  velocity  of  the  entering 
water  is  12  ft.  per  second.     The  centre  of  the  sluice-opening  is  -54ft. 
above  the  point  of  entrance,  and  the  width  of  the  sluice  is  3!  ft.     The 
wheel  has  48  buckets.     The  distance  between  the  wheel  and  breast  is  % 
inch.     The  bucket  passes  through  .9  ft.  while  receiving  water,  and  the 
depth  of  the  water-surface  in  the  bucket  below  the  point  of  entrance  is 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS. 

1.25  ft.  Find  (a)  the  angular  distance  of  the  point  of  entrance  from  the 
horizontal,  (b)  the  fall  in  the  breast,  (<r)  the  head  of  water  over  the 
sluice,  (d)  the  velocity  of  the  water  in  the  bucket  the  moment  entrance 
ceases,  (e)  the  total  mechanical  effect,  disregarding  axle  friction. 

Ans.  (a)  53°  53';  (b)  6.525  ft.;  (c)  1.935  ft.;  (d)  14.9  ft.;  (e)  15.59  H.P. 

61.  In  the  preceding  question,  if  the  energy  absorbed  by  axle  fric- 
tion, etc.,  is  743  ft.-lbs.,  find  the  efficiency  of  the  wheel.  Ans.  f. 

62.  20  cub.  ft.  of  water  per  second  enter  an  undershot  wheel  of  20  ft. 
diameter  in  a  straight  race,  the  fall  being  3  ft.     The  depth  of  the  enter- 
ing stream  is  £  ft.     The  width  of  the  wheel  is  4f  ft.,  and  the  clearance  is 
f  inch.     The  number  of  the  floats,  of  which  four  are  immersed,  is  48, 
and  each  is  i  ft.  long.     The  weight  of  the  wheel  is  7200  Ibs.,  the  radius 
of  the  axle  is  if  in.,  and  the  coefficient  of  friction  is  .1.     Find  (a)  the 
best  speed  for  the  wheel,  (b}  the  corresponding  mechanical  effect,  (c)  the 
efficiency. 

Ans.  (a)  6  ft.  per  second  ;  (ff)  2.32  H.P.,  assuming  the  speed  of 
wheel  reduced  to  5.74  ft.  per  second  by  axle  friction  ;  (c}  .34. 

63.  A  downward-flow  turbine  of  24  in.  internal  diameter  passes  10 
cub.  ft.  of  water  per  second  under  a  head  of  31  ft ;  the  depth  of  the 
wheel  is   i   ft.    and   its  width  6  in.     Find  the  efficiency,  assuming  the 
whirling  velocity  at  outlet  to  be  nil.  Ans.  .997. 

64.  A  downward-flow  turbine  of  5   ft.  external  diameter  passes  20 
cub.   ft.  of  water    per  second  under  a  head  of  4  ft.,  the  depth  of  the 
wheel  being  5  ft.     The  water  enters  the  wheel  at  an  angle  of  60°  with 
the  vertical,  the  receiving-lip  of  the  wheel-vanes  is  vertical,  and  the  ve- 
locity of  whirl  at  outlet  is  nil.     Find  the  internal  diameter  and  the 
speed  in  revolutions  per  minute.  Ans.  4.68  ft.;  46.53. 

65.  A  downward-flow  turbine  has  an  internal  diameter  of  24  in.;  the 
breadth  of  the  wheel  is  6  in.;  the  turbine  passes  33  cub.  ft.  per  second 
under  an  effective  head  of  16  ft.    Assuming  the  whirling  velocity  at  out- 
let to  be  nil,  find  the  efficiency  and  power  of  the  turbine.     If  the  vane- 
lip  at  inlet  is  radial,  finci  the  direction  of  the  vane  at  outlet,  and  the 
speed  of  the  turbine  in  revolutions  per  minute. 

Ans.  .931  ;  55.865  H.P.;  ft  =  y  =  21°  2' ;  166.7. 

66.  Discuss   the   preceding   question    on   the   assumption   that   the 
peripheral  speed  at  outlet  (»9)  is  equal  to  the  speed  of  the  water  at  that 
point  relatively  to  the  wheel  (  F»). 

Ans.  .928  ;  55.715  H.P.;  /3  =  21°  47'  and  y  =  20°  21'. 

67.  An  axial-flow  impulse  turbine  of  5  ft.  mean  diameter  passes  170 
cub.  ft.  of  water  per  second  under  an  effective  head  of  8.6  ft.;  the  depth 
of  the  wheel  is  .9  ft.     At  what  angle  should  the  water  enter  the  wheel  to 
give  an  efficiency  of  81  per  cent,  the  width  of  the  wheel  being  constant 
and  disregarding  hydraulic  resistances  ?  Ans.  =27    16'. 

68.  In  example  67,  find  (a)  the  velocity  with  which  the  water  enters 


324  HYDRA  ULICS. 

the  wheel  (b)  the  speed  of  the  turbine  in  revolutions  per  minute,  (c)  the 
directions  of  the  vane  edges  at  inlet  and  outlet,  (d)  the  velocity  of  the 
water  as  it  leaves  the  wheel,  (e)  the  power  of  the  turbine. 

Ans.  (a)  2346ft.  per  sec.;  (b)  45.08  ;  (c)  a  =  130°  05',  ft  =  42°  19'; 
(d}  10.748  ft.  per  sec.;  (e)  148.65  H.P. 

69.  In  example  67,  if  instead  of  assuming  that  the  whirling  velocity 
at  exit  is  nil,  it  is  assumed  that  the  peripheral  speed  (u*)  of  the  wheel  at 
the  mean  radius  is  equal  to  the  relative  velocity  ( F2)  of  the  water  at  exit, 
show  how  the  several  results  are  affected. 

Ans.  y  =  25°  6';  (a)  23.46  ft.  per  second  ;  (b)  54.638  ; 
(c)  a=  124°  54',  0  =  44°  7' ;  (d)  10.748  ft.  per  second  ; 
(e)  148.65  H.P. 

70.  In  examples  68  and  69,  assuming  that  the  hydraulic  resistances 
necessitate  an  increase  of  i2£  per  cent  in  the  head  equivalent  to  the  ve- 
locity with  which  the  water  enters  the  wheel,  and  an  increase  of  10  per 
cent  in  the  head  equivalent  to  the  relative  velocity  ( VJ)  at  outlet,  show 
how  the  several  results  are  affected. 

Ans.  Question  68.  (a)  22.12  ft.  per  sec.;  (b)  47.82; 

(c)  a  —  121°  30',  ft  =  40°  9';  (d)  10.748  ft.  per  sec.; 
(e)  148.65  H.P. 
Question  69.  (a)  22.119  ft.  per  sec.;  (b}  50.97; 

(c)  a=  124°  91',  ft  =47°  28';  (d}  10.748  ft.  per  sec.; 
(<?)  148.65  H.P. 

71.  The  efficiency  of  an  axial -flow  turbine  is  90  per  cent,  and  it  passes 
12  cub.  ft.  per  second  under  an  effective  head  of  40  ft.    At  the  mean 
radius  the  water  enters  at  an  angle  of  30°  with  the  wheel's  face,  and  the 
whirling  velocity  at  outlet  is  nil.     Find  (a)  the  velocity  with  which  the 
water  enters  and  leaves  the  wheel,  (b)  the  directions  of  the  vane  at  inlet 
and  outlet,  (c)  the  sectional  areas  of  the  inlet-  and  outlet-orifices,  (d}  the 
speed  of  the  wheel  in  revolutions  per  minute,  (<?)  the  power  of  the  tur- 
bine. Ans.  (a)  42%  ft.  per  second  ;  16  ft.  per  second  ; 

(b)  a  =  49°  6',  0  =  2i°  3'; 

(c)  .75  sq.ft.; 

W). I9&39;  (<f)49&  H.P. 

72.  An  axial-flow  turbine  of  5  ft.  mean  radius  passes  212  cub.  ft.  of 
water  per  second  under  a  total  effective  head  of  12.1  ft.     At  the  mean 
radius,  the  direction  of  the  inflowing  water  makes  an  angle  of  70°  with 
the  vertical,  and  the  vane-lip  at  the  outlet  makes  an  angle  of  17°  with 
the  wheel's  periphery.     If  the  whirling  velocity  at  the  outlet-surface  is 
nil,  find  (a)  the  velocity  with  which  the  water  must  enter  the  wheel  to 
give  an  efficiency  of  .953  per  cent.     Also  find  (b)  the  direction  of  the 
vane-lip  at  outlet,  (c)  the  speed  of  the  wheel  in  revolutions  per  minute, 
(d}  the  widths  and  areas  of  the  inlet-  and  outlet-orifices,  (e)  the  power 
of  the  turbine. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    $2$ 

Ans.  (a)  19.9  ft.  per  sec.; 

(6)  «=8i°22';  (*)  37.67; 

(d)  .991  ft.;  31.148  sq.ft.;  1.181  ft;  35.14  sq.ft.; 

(e)  277- 799  H.  P. 

73.  An  axial-flow  impulse  turbine  passes    170  cub.  ft.  of  water  per 
second  under  an  effective  head  of  9.5  ft.,  the  depth  of  the  wheel  being  .9 
ft.  and  its  mean  radius  4.2  ft.    The  vane-lip  at  the  outlet  makes  an  angle 
of  72°  with  the  vertical.     Assuming  that  the  whole  of  the  effective  head 
is  transformed  into  useful  work,  and  that  the  whirling  velocity  at  the 
outlet-surface  is  nil,  find  (a)  the  inclination  to  the  vertical  of  the  outlet- 
lip  of  the  guide-vane,  (b)  the  direction  of  the  inlet-lip  of  the  wheel-vane, 
(c)  the  efficiency ;  first  neglecting  hydraulic  resistances,  and  second  taking 
these  resistances  into  account. 

Ans.   First.        (a)  59°  52' ;     (b)  60°  16' ;  (c)  .905; 
Second,  (a)  52°  52';    (b)  74°  16' ;  (c)  .804. 

74.  In  the  preceding  example  find  the  inlet-  and  outlet-orifice  areas 
in  the  two  cases.  Ans.  First.       8.12  sq.  ft.  ;  22.4  sq.  ft. ; 

Second.  9.64  sq.  ft.;  28.56  sq.  ft. 

75.  An  axial-flow  turbine   passes   200  cub.  ft.  of  water  per  second 
under  a  head   of  14  ft.,  the  depth  of  the  wheel  being  i  ft.     The  mean 
radius  of  the  wheel  is  3  ft.  ;  the  areas  of  the  inlet-  and  outlet-surfaces  are 
in  the  ratio  of  7  to  8 ;  the  water  enters  the  wheel  at  an  angle  of  21°  to 
the  wheel  face,  and  the  outlet  edge  of  the  vane  makes  an  angle  of  16° 
with  the  face.    Find  the  speed,  efficiency,  and  power  of  the  turbine,  and 
also  the  direction  of  the  inlet-lip  of  the  vanes. 

Ans.  73.69  revolutions  per  minute;  .954;  325.24311.?.; 
a  =65°  57'. 

76.  In   question    11,   if  there  are  62  wheel,  and  66  guide-vanes,  the 
thickness  of  the  latter  being  .2  in.  and  of  the  former  .4  in.,  find  the 
width  of  the  inlet-orifices. 

77.  Water  is  delivered  to  an  O.  F.  turbine  at  a  radius  of  24  in.  with  a 
whirling  velocity  of  20  ft.  per  second,  and  leaves  in  a  reverse  direction 
at  a  radius  of  4  ft.  with  a  whirling  velocity  of  10  ft.  per  second.     If  the 
linear  velocity  of  the  inlet-surface  is  20  ft.  per  second,  find  the  head 
equivalent  to  the  work  done  in  driving  the  wheel.  Ans.  24.8  ft. 

78.  An  outward-flow  turbine  of  9.5  in.  external  diameter  works  under 
an  effective  head  of  270  ft.     Find  the  speed  in  revolutions  per  minute, 
assuming  that  the  whirling  velocity  at  the  inlet-surface  relatively  to  the 
wheel  is  nil  and  that  the  efficiency  is  unity.  Ans.  2242. 

79.  An  outward-flow  turbine,  whose  external  and  internal  diameters 
are  8  ft.  and  5^  ft.  respectively,  makes  26  revolutions  per  minute  under 
an  effective  head  of  4  ft.     The  water  enters  the  wheel  in  a  direction 
making  an  angle  of  36°  (y)  with  the  direction  of  motion  at  the  point  of 
entrance.    Determine  the  angles  of  the  moving  vane  at  ingress  and  egress, 


HYDRAULICS. 

the  efficiency  being  .85.   Also  find  the  energy  per  pound  of  water  carried 
away  by  the  water  as  it  leaves  the  turbine. 

Ans.  a=  129°  59',  fi=  29°  38';  .6  ft.-lbs. 

80.  Construct  an  outward-flow  turbine  from  the  following  data :  the 
fall  =  5  ft.  ;  internal  diameter  =  1.8  ft.;  external  diameter  =  2.45    ft.; 
quantity  of  water  passed  per  second  =  30  cub.  it.,y  =  30° ;  efficiency  =  .9. 
Ans.  a  =  108°  15';  fi  =  21°  3';  Ai  =3.897  sq.  ft. ;  A*  =  5.303; 

d\  =  d*  =  .688  ft.,  neglecting  thickness  of  vanes. 
Si  Assuming  that  the  intensities  of  the  pressure  at  the  receiving 
and  discharging  edges  of  the  moving  vanes  of  a  Fourneyron  turbine  are 
equal,  and  also  that  the  rim  velocity  and  the  velocity  of  the  water 
relatively  to  the  wheel  at  the  discharging-surface  are  equal,  show  that 
the  direction  of  the  impinging  stream  must  bisect  the  angle  between 
the  direction  of  motion  and  the  tangent  to  the  vane  at  the  receiving 

r?       sin  /? 

edge.    Also  show  that  — „  =  -. . 

r-i        sin  ?.y 

82.  If  the  areas  of  the  inlet-  and  outlet-orifices  of  an    inward-  or 
outward-flow  impulse  turbine  are  equal,  show  that  the  efficiency  of  the 
turbine  is  cos2  y ,  y  being  the  angle  which  the  direction  of  the  entering 
water  makes  with  the  wheel's  periphery. 

83.  A  radial  impulse  turbine  of  4.5  ft.  and  4  ft.  external  and  internal 
radii  passes  8^  cub.  ft.    of  water  per  second   under  an  effective   head 
of  560  ft.     The  direction  of  the  entering  water  is  inclined  at  17°  to  the 
wheel's  periphery,  and  the  wheel  has  the  same  depth  at  the  inlet-  and 
outlet-surfaces.     If  the  peripheral  speed  at  the  outlet-surfe  (w2)  is  equal 
to  the  relative  velocity  of  the  water  (F2)  with  respect  to  the  wheel,  find 
(a)  the  efficiency,  (b]  the  speed  of  the  turbine  in  revolutions  per  minute, 
(c)  the  sectional  areas  of  the  stream  at  inlet  and  outlet,  (d)  the  direction 
of  the  vane-outlet  edge,  (e)  the  velocity  of  the  water  as  it  leaves  the 
wheel,  (/")  the  power  of  the  turbine. 

Ans.   (a)  .873;     (£)   209.94;     (c)   .15357  sq.    ft.;    .13651    sq.    ft.; 
(d)  ft  =  45°  2' ;  (e)  67.39  ft.  per  sec.;  (f)  472.33  H.P. 

84.  In    the   preceding    question    examine   how   the   results  will   be 
affected  when  hydraulic  resistances  are  taken  into  account,  allowing  .94 
as  a  coefficient  of  velocity  for  the  water  on  entering  the  wheel,  and 
assuming  that  the   head  equivalent  to   the   relative  velocity  ( Fa)   on 
leaving  the  wheel  is  increased  by  10  per  cent. 

Ans.  (a)  .886;    (£)  193.76  revols.  per  minute; 

(c)  .163  sq.  ft.;  .145  sq.  ft.; 

(d)  ft  =  46°  1 8';    (*)  63.842  ft.  per  sec.; 
(/)  479-39  H.P. 

85.  A  radial  outward-flow  turbine  of  the  impulse  type  passes  8^  cub. 
ft.  of  water  per  minute  under  an  effective  head  of  560  ft.;  the  width  of 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    327 

the  wheel  is  7^  in.;  the  radius  to  the  outlet-surface  is  1.15  times  the 
radius  to  the  inlet-surface  ;  the  linear  velocity  of  the  inlet-surface  is  87 
ft.  per  second  ;  the  direction  of  the  water  at  entrance  makes  an  angle  of 
17°  with  the  wheel's  periphery.  Find  (a)  the  efficiency,  (b}  the  lip- 
angles,  (c)  the  areas  of  the  inlet-  and  outlet-orifices,  neglecting  first 
hydraulic  resistances,  and  second  taking  these  resistances  into  account. 
Ans.  First,  (a)  .878;  (b)  a  =  149°  31'  and  ft  =  33°  20'; 

(c)  .1535  sq.  ft.  and  .1283  sq.  ft.; 
Second,  (a)  .826;  (b)  a  —  i48°and  ft  =  17°  19' ; 
(c)  .183  sq.  ft.  and  .306  sq.  ft. 

86.  Construct  a  Fourneyron  turbine  for  a  fall  of  5  ft.  with  30  cub.  ft. 

of  water  per  second,  a.  =  80°,  y  =  30°,   —  =  1.35.       Assume    u%  =  Fa, 

and  neglect  hydraulic  resistances. 

Ans.  ft  =  16°  42' ;  Ai  =  4.29  sq.  ft.;  A*  =  5.8189  ft. ;  rj  =  .915 ; 
if  ri  —  1.8  ft.,  then  di  =  d*  =  .38  ft. 

87.  In    an    inward-flow  turbine    passing  400  gallons   of  water  per 
minute,  the  slope  of  the  guide-vane  lips  is  i  in  5,  the  radii  to  the  inlet- 
and  outlet-surfaces  are  i  ft.  and  6  ins.,  respectively;  the  breadth  of  the 
inlet-orifices  is  1.25  ft.     Find  the  efficiency.  Ans.  .98. 

88.  An   I.  F.  turbine,  of  4   ft.  external  diameter,  works   under  an 
effective  head  of  250  ft.     Find  the  speed  of  the  wheel  in  revolutions  per 
minute.  Ans.  427. 

89.  An   I.  F.  turbine  of  4  ft.  external  and  3  ft.  internal  diameter, 
makes  360  revolutions  per  minute.     The  sectional  area  of  flow  is  3  sq. 
ft.  and  is  the  same  in  every  part  of  the  turbine.     The  direction  of  the 
inflowing  water  makes  an   angle   of   30°   with  the  wheel's   periphery. 
Assuming  that  the  whirling  velocity  at  the  outlet-surface  is  nil,  find 
(a)  the  efficiency,  (b)  the  H.P.,  and  (c)  the  delivery  in  cubic  feet  per 
minute.     The  total  head  is  200  ft.  Ans.  (a)  .86 ;  (<£)  2476.8  ;  (c)  7593. 

90.  An  inward-flow  turbine  being  required  for  an  available  head  of 
20  ft.  and  a  discharge  of  800  cub.  ft.  per  minute,  determine  (a)  the  size 
and  (b)  the  speed  of  the  wheel,  (c)  the  inclinations   of  the  guide  and 
wheel-vanes,  and  (d}  the  efficiency  of  the  turbine,  assuming  r2  =  \r\  =• 
depth  of  wheel ;  vr'  =  ^  \/2g-ff;  v^,"  =  o  and  a.  =  90°. 

Ans.  (a)  r2  =  .487  ft.,  ri  =  .974  ft.; 

(b)  240  revolutions  per  minute; 

(c)  y  =  10°  21',   ft  =  36°  8' ;  (d)  93!  per  cent. 

91.  A  vortex  turbine  passes  Q  cub.  ft.  of  water  per  second  under  an 
effective  head  of  H  ft.      The  inlet-lip  of  the  vanes  is  radial  and  the 
direction  of  the  entering  water  makes  an  angle  of  30°  with  the  wheel's 

periphery.    The  areas  of  the  inlet-  and  outlet-orifices  are  —   —  and  - 

o  C 


328  HYDRA  ULICS. 

respectively,  and  the  width  of  the  wheel  is  — ,  D  being  the  diameter  of 

the  inlet-surface.     If  the  whirling  velocity  at  the  outlet-surface  is  nil, 
find  (a)  the  efficiency,  (b)  the  direction  of  the  outlet  edge  of  the  vane, 

(c)  the  velocity  with  which  the  water  enters  and  leaves  the  wheel ; 

(d)  the  speed  of  the  wheel  in  revolutions  per  minute,    (<?)  the  diameters 
of  the  inlet-  and  outlet-surfaces. 

Ans.  (a)  .938;     (b)  /3=24°  17';    (V)  6.3291^  ;    1.977 HI  \ 

(d)  "6.7^;   (*)  -896 j|;  .yijrfj. 

92.  A  vortex  turbine  passes  1 1  cub.  ft.  of  water  per  second  under 
a  head  of  35  -ft.;   the  diameter   of  the   outlet-surface    is  2  ft.  and  its 
breadth  6  in.     Find  the  power  of  the  turbine,  disregarding  friction  and 
assuming  that  the  whirling  velocity  at  the  outlet-surface  is  nil. 

Ans.  43.5  H.P. 

93.  An   inward-flow  turbine  has  an  internal  radius  of  12  in.  and  an 
external  radius  of  24  in.;  the  water  enters  at  15°  with  the  tangent  to  the 
circumference,  and  is  discharged  radially;  the  velocity  of  outer  periph- 
ery of  wheel  is   16  ft.  per  second.     Find  the  angles  of  the  vanes  at  the 
inner  and  outer  circumferences,  and  the  useful  work  done  per  pound  of 
fluid.  Ans.  ft  =  32°,  a  =  118°  I';  9.33  ft.-lbs. 

94.  A  radial  impulse  turbine  passes  S^  cub.  ft.  of  water  under  an 
effective  head  of  560  ft.     The  direction  of  the  entering  water  is  inclined 
at  17°  to  the  wheel's  periphery.     The  linear  speed  of  the  inlet-surface  is 
87  ft.  per  second.     Assuming  that  the  velocity  of  whirl  at  the  outlet  is 
nil,  and  disregarding  hydraulic  resistances,  find  (a)  the  efficiency,  (b)  the 
velocity  with  which  the  water  enters  the  wheel,  (c)  the  velocity  of  the 
water  as  it  leaves  the  wheel,  (d)  the  sectional  areas  of  the  inflowing  and 
outflowing   stream,  (<?)   the  direction  of  the  vane-lip  at  inlet,  (/)  the 
power  of  the  turbine. 

The  radii  of  the  inlet-  and  outlet-surfaces  are  4^|  ft.  and  4$  ft.  respect- 
ively.    Find  (g)  the  direction  of  the  vane  edge  at  outlet. 

Ans.  (a)  .879;  (b)  189.31  ft.  per  sec.  ;  (c)  65.86  ft.  per  sec.; 
(d}  .15356  sq.  ft.  ;  .129  sq.  ft.;  (e)  a  =  149°  34'; 
(7)47543  H.P.  ;  (g)  ft  =  33°  21'. 

95.  In  the  preceding  example  show  how  the  results  are  affected  by 
taking  .94  as  the  coefficient  of  velocity  in  calculating  the  velocity  with 
which  the  water  enters  the  wheel. 

Ans.  (a)  .828;  (b)  178.49  ft.  per  sec.;  (c)  78.36  ft.  per  sec.  ; 
(d)  .163  sq.  ft.  ;  .1085  sq.  ft.;  (e}  a  =  148°  3' ; 
(/) .441.25  H.P.;  (£•)/*  =  38"  4'* 

96.  In  an  I.  F.  turbine  the  radius  to  the  inlet-surface  is  twice  that  to 
the  outlet-surface ;  the  linear  velocity  of  the  inlet-surface  is  one  half 
that  due  to  the  head ;  the  water  enters  the  wheel  with  a  velocity  of  flow 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    329 

{vr')  equal  to  one  eighth  that  due  to  the  head,  and  the  sectional  area  of 
the  water-way  is  constant  from  inlet  to  outlet.  Find  the  angle  between 
the  discharging-lip  of  the  vane  and  the  wheel's  periphery,  the  whirling 
velocity  at  the  outlet-surface  being  nil.  Ans.  Cot"1  4/8. 

97.  In  a  vortex  turbine  the  depth  of  the  inlet-orifices  is  one  eighth  of 

the  diameter  of  the  wheel  (=  ~]  and  —  of  the  depth  of  the  outlet- 

\        8  /  32 

orifices.     The  width  of  the  wheel  is  one  tenth  of  the  diameter  (  =  — -). 

\      io/ 

The  inlet-lip  of  the  vanes  is  radial,  and  the  water  enters  at  an  angle  of 
30°  with  the  inlet  periphery.  Find  the  size,  speed,  and  efficiency  of  the 
turbine  in  terms  of  the  supply  of  water  Q  and  the  effective  head  H. 
Also  find  the  direction  of  the  outlet  edge  of  the  vanes. 

Q^ 
Ans.  I.  Assume  vw"  =  o.     Then  r\  =  .448 — -,  ; 

H* 
No.  of  revolutions  per  minute  =  1 16.7 — r  ; 

r;  =  .938;  /3  =  24°  17'. 

Qk 

II.  Assume  w2  =  F2.    Then  r\  —  .45 — -  ; 

H 4 

H* 
No.  of  revolutions  per  minute  ==  116. -3 — r 

Qk 

77  =  .935;  $  =  26°  49'. 

98.  A  vortex  turbine  with  a  wheel  of  2  ft.  diameter  and  6  in.  breadth 
passes  io  cubic  ft.  of  water  per  second  under  a  head  of  32  feet.     Find 
the  inclination  of  the  guides  and  the  power  of  the  turbine.     Assume 
as  =  V?.  and  a  =  90°.  Ans.  5°  41',  36T4T  H.P. 

99.  An  inward-flow  turbine  has  an  internal  radius  of  12  in.  and  an 
external  radius  of  24  in. ;  the  water  enters  at  15°  with  the  tangent  to  the 
circumference,  and  is  discharged  radially;  the  velocity  of  radial  flow  is 
5  ft.  at  botn  circumferences;  the  velocity  of  outer  periphery  of  wheel  is 
16  ft.  per  second.     Find  the  angles  of  the  vanes  at  the  inner  and  outer 
circumferences,  and  the  useful  work  done  per  pound  of  fluid. 

Ans.   ft  =  32° ;  d  =  1 18°  i' ;  9.33  ft.-lbs. 

100.  A  radial   I.  F.  reaction   turbine,  with  or  without  draught-pipe, 
passes  113  cub.  ft.  of  water  under  an  effective  head  of  13  ft.     The  radius 
to  the  inlet-surface  is  1.169  times  the  radius  to  the  outlet- surface,  and  the 
ratio  of  the  outlet  to  the  inlet  area  is  .92.     The  vane-lip  at  outlet  makes 
an  angle  of  15°  with  the  wheel's  periphery,  and  the  water  enters  at  an 
angle  of  1 2°  with  the  wheel's  periphery.    The  sectional  area  of  the  draught- 
tube  (if  there  is  one)  at  the  point  of  discharge  is  1.035  times  the  sectional 
area  of  the  outlet-orifice.     Show  that  the  effective  work  per  pound  of 


33°  HYDRA  ULICS. 

water  is  13  ft.-lbs.,  and  that  the  work  consumed  in  hydraulic  resistance 
(Art.  28,  page  303)  is  nearly  1.96  ft.-lbs.  ;  also  find  Ai,  A^  z/a,  and  the 
efficiency. 

Ans.  (a)  28.26  sq.  ft.  ;  26.12  sq.  ft.  ;  (b)  4.4  ft.  per  sec. ;  .977. 

101.  In  the  preceding  example,  if  the  radius  to  the  inlet-surface  is  4 
ft.,  find  (a)  the  speed  of  the  wheel  in  revolutions  per  minute.     Also  find 
(b)  the  depth  of  the  wheel  at  inlet  and  outlet,  the  guide-vanes  being  40 
and  the  wheel-vanes  41  in  number,  and  the  thickness  of  the  former  being 
T\  inch  and  of  the  latter  J  inch.        Ans.  (a)  38.95  ;  (b)  1.23  ft. ;  1.32  ft. 

102.  In  example  38  find  the  efficiency  if  the  diameter  of  the  draught- 
tube  is  made  the  same  as  the  diameter  of  the  outlet-surface,  the  lower 
edge  of  the  tube  being  rounded.     What  will  be  the  "loss  in  shock"  in 
the  tube  per  pound  of  water?  Ans.  .861  ;  .071  ft.-lbs. 

103.  An  axial-flow  turbine  is  to  be  used  for  raising  water.     Explain 
how  the  vanes  should  be  arranged,  write  down  the  resulting  equations, 
and  determine  the  efficiency. 

104.  Write  down  the  equations  for  a  Jonval  modification  of  Euler's 
turbine. 

105.  An  inward-flow  turbine  has  an  external  diameter  of  3  ft.  and  an 
internal  diameter  of  2  ft.     It   passes   12  cub.   ft.  of  water  per  second 
under  an  effective  head  of  40  ft.     The  water  enters  the  wheel  at  an 
angle  of  30°  with  the  wheel's  periphery,  and  the  depth  of  the  outlet- 
orifices  is  twice  the  depth  of  the  inlet-orifices.     The  efficiency  of  the 
turbine  is  .9.     Disregarding  friction,  find  (a)  the  vane-angles  at  inlet  and 
outlet,  (b)  the  velocity  with  which  the  water  leaves  the  wheel,  (c)  the 
speed  of  the  turbine  in  revolutions  per  minute,  (d)  the  velocity  with 
which  the  water  enters  the  wheel,  (e)  the  areas  of  the  outlet-  and  inlet- 
orifices,  (/)  the  power  of  the  turbine. 

Ans.  (d)  a  =  105.09',  ft  =  35°  35'  (b)  16  ft.  per  sec.  ;  (c)  198.39; 

(d)  42!  ft.  per  sec.  ;  (e)  .5625  sq.  ft.  ;  .75  sq.  ft. ;  (/)  49^  H.P. 

106.  A  centrifugal  pump  with  a  12-in.  fan  delivers  1000  gallons  per 
minute,  the  actual  lift  being  20  ft.  and  the  gross  lift  (allowing  for  fric- 
tion, etc.)  30  ft.     The  velocity  of  whirl  at  the  outlet-surface  is  reduced 
one  half.    Find  the  revolutions  of  the  pump  per  minute.       Ans.  623.1. 

107.  In  a  centrifugal  pump  the  external  diameter  of  the  fan  is  2  ft.,  the 
internal  i  ft.,  and  the  width  6  in.     Determine  the  speed  and  efficiency 
of  the  pump  when  delivering  2000  cub.  ft.  per  minute  against  a  pressure 
head   of  64  ft.,   the   inclination  of  the  wheel-vanes   at   outlet-surface 
being  90°.  Ans.  643.36  revols.  per  min. ;  .656. 

108.  A  centrifugal  pump  delivers  1500  gallons  per  minute.  Fan,  16  in. 
diameter;  lift,  25  ft. ;  inclination  of  vanes  at  outer  periphery  to  the  tan- 
gent, 30°.     Find  the  breadth  at  the  outer  periphery  that  the  velocity  of 
whirl  may  be  reduced  one  half,  and  also  the  rev6lutions  per  minute,  as- 
suming iht  gross  lift  to  be  i£  times  the  actual  lift. 


HYDRAULIC  MOTORS  AND    CENTRIFUGAL   PUMPS.    331 

Also  find  the  proper  sectional  area  of  the  chamber  surrounding  the  fan 
for  the  proposed  delivery  and  lift.  Examine  the  working  of  the  pump  at 
a  lift  of  15  ft.  Ans.  Breadth,  f  in.;  revolutions,  700;  24  sq.  in. 

109.  For  a  given  discharge  (0  and  head  (77),  and  considering  only  the 
losses  of  head  due  to  flow  and  to  the  resistance  in  the  wheel,  show  that 
the  maximum  efficiency  of  a  centrifugal  pump  of  chamber  D  is 


A  being  a  constant  depending  on  the  size  of  the  wheel. 

no.  A  centrifugal  pump  lifts  35  cub.  ft.  of  water  per  second  a  height 
of  20  ft.  At  the  outer  periphery  the  vane-angle  (ft)  is  15°  and  the  radial 
velocity  is  5  ft.  per  second.  If  the  wheel  makes  140  revolutions  per 
minute,  find  (a}  its  diameter.  If  the  diameter  of  the  outer  periphery  of 
the  wheel  is  three  times  that  of  the  inner  periphery  and  if  the  radial 
velocity  at  the  latter  is  8  ft.  per  second,  find  (b)  the  vane-angle  at  the  in- 
ner periphery  and  (c)  the  depths  of  the  wheel  at  the  inner  and  outer 
peripheries.  .  Ans.  (a)  5^  ft.  ;  (b)  30°  58';  (c)  n.i  in.;  5.76  in. 

in.  The  pump  in  the  preceding  example  is  supplied  with  a  vortex 
chamber  of  6£  ft.  diameter.  Show  that  the  "gain  of  head"  is  a  maxi- 
mum when  the  velocity  of  flow  in  the  volute  is  8.46  ft.  per  second.  Also 
show  that  the  frictional  loss  of  head  is  4.1785  ft. 

112.  In  a  centrifugal  pump  the  diameter  of  the  fan  =  16  in.  ,  the  depth 
=  2  in.,  the  lift  =  25  ft.,  and  the  delivery  =  300  cub.  ft.  per  minute. 
Determine  (a)  the  speed,  (b)  the  efficiency,  and  (c)  the  power  expended 
when  the  vane-angle  (ft)  at  the  outer  periphery  is  (i)  90°,  (2)  45°,  and  (3) 
30°.  Ans.  (i)  (a)  785     revols.  per  min.  ;  (b)  .47  ;  (c)  30     H.P.  ; 

(2)  (a)  805.8       ••         "        "        (b)  .58  ;  (f)  24.4  H.P.  ; 

(3)  (a)  846.1       "         ««       "       (b)  .68  ;  (c)  22.9  H.P. 

113.  An  Appold  pump  delivers  10,000  gallons  per  minute.  The  gross 
lift  is  50  ft.     The  radial  velocity  at  the  outlet-surface  is  one  eighth  of 
that  due  to  the.  gross  lift,  and  the  velocity  of  whirl  and  the  peripheral 
velocity  are  reduced  one  half.     Find  (a)  the  radius  of  the  wheel,  (b)  the 
vane-  angles,  (c}  the  speed  of  the  wheel,  (d)  the  efficiency. 

Take  the  breadth  of  the  wheel  at  outlet  equal  to  one  sixth  of  the  ra- 
dius, and^-  =  32. 

Ans.  (a)  1.9  ft.  ;  (b)  56°  16'  ;  23°  16;  (c)  331  revols  per  min.  ;  (d)  .74. 

114.  The  internal  and  external  diameters  of  the  fan  of  a  centrifugal 
pump  are  9  in.  and  18  in.,  respectively  ;  the  depth  is  6  in.,  and  it  passes 
400  cub.  ft.  per  minute  against  a  pressure  head  of  16  ft.     The  inclination 
(ft)  of  the  discharging-lips  of  the  fan  being  30°,  determine  (a)  the  speed, 
(b)  the  efficiency,  (c)  the  power  expended,  and  (d)  the  inclination  of  the 
receiving-lips  of  the  fan. 

Ans.  (0)413.58  revols.  per  min.  ;  (b)  .571  ;  (c)  21.23  H.P.  ;  (d)  19°  48'. 


3  3  2  HYDRA  ULICS. 

Find  the  efficiency  when  a  vortex  chamber  36  in.  in  diameter  Sur- 
rounds the  fan.  Ans.  .581. 

115.  A  centrifugal  pump  with  a  gross  lift  of  17  ft.  delivers  25  cub.  ft. 
of  water  per  second.     At  the  outer  periphery  the  vane-angle  is  80°  and 
the  radial  velocity  is  5  ft.  per  second.     The  diameters  of  the  outer  and 
inner  peripheries  of  the  disc  are  54  in.  and  18  in.,  respectively,  and  the 
hydraulic  efficiency  is  .75.     Find  (a)  the  speed  of  the  fan,  (b)  the  vane 
angle  at   the   outlet   periphery,  (c)  the  velocity  of  flow  in  the  volute, 
(d)  the  diameter  of  the  volute,  (<?)  the  diameter  of  the  suction-pipe. 

If  there  are  six  £-in.  vanes,  find  (/)  the  width  of  the  disc  at  the  outer 
and  inner  peripheries. 

Assuming  the  discharge-pipe  to  be  4  ft.  per  second,  show  that  there  is 
a  loss  of  5.026  ft.  of  head  due  to  hydraulic  friction. 

Ans.  (a)  116  revols.  per  min.  ;  (b)  41°  14' ;  (c)  26.0  ft.  per  second  ; 
(d)  14.7  in.  ;  (e)  33.8  in.  ;  (/)  9.64  in. ;  4.8  in. 

116.  The  vane  of  a  centrifugal  pump  or  turbine  is  the  involute  of  a 
circle  concentric  with  the  pump  circumference.    Show  that  V\  =  V*  in  an 

I.  F.  or  O.  F.,  and  '£  =  -  in  a  D.  F. 

Vt       rz 

117.  A  race  is  straight  and  close  fitting  so  that  the  loss  of  effect  due 
to  escape  of  water  may  be  disregarded.     A  single  undershot  wheel  with 
plane  floats  is  replaced  by  four  similar  tandem  wheels.     If  the  delivery 
of  each  of  the  four  wheels  is  the  same,  and  if  it  is  assumed  that  the 
water  reaches  each  wheel  with  the  same  velocity  with  which  it  leaves  the 
preceding  wheel,  find  the  total  maximum  velocity  due  to  impact. 

Ans.  i£  times  the  delivery  of  the  single  wheel. 

118.  Discuss  the  preceding  example,  assuming  that  the  delivery  of 
each  wheel  is  not  the  same,  but  that  the  total  delivery  is  a. maximum. 

Ans.  1.6  times  the  delivery  of  the  single  wheel. 

119.  If  n  wheels  of  the  same  type  are  substituted  for  the  single  wheel 
in  example  117,  and  if  the  assumptions  are  the  same  as  those  in  example 
1 1 8,  show  that  the  total  delivery  of  the  n  wheels  is  to  the  delivery  of  the 
single,  wheel  in  the  ratio  of  2«  to  in  +  i,  and  that,  theoretically,  if  the 
number  is  made  very  large,  they  will  approximately  give  the  entire  work 
of  the  fall. 


INDEX. 


Abbott,  148,  151 

Abrupt   changes    of  section,    loss    of 

head  due  to,  89 
Accumulator,  215 
Air  in  a  pipe,  no 
Aqueducts,  flow  in,  142 
Arc  of   discharge  in   overshot  wheel, 

256 
Axial-flow  turbine,  282 

Barker's  mill,  272 

Barlow  curve,  47,  50 

Barometer,  water,  5 

Bazin,  145,  152,  154,  166 

Bazin's  velocity  curve  and  formula,  154 

Beard  more,  134 

Beaufoy,  7 

Bends,  loss  of  head  due  to,  92 

Bernouilli's  theorem,  6 

"        applications  of,  9 

Bidone,  23,  36,  166 

Boileau,  153 

Boileau's  velocity  curve  and  formula, 
154 

Borda,  36 

Borda's  mouthpiece,  34 

Bossut,  227 

Bovey's  tables   of  coefficients  of  dis- 
charge, 24,  25 

Boyden's  hook  gauge,  173 

Branched  pipe  connecting  three  reser- 
voirs, in 

Branch  main  of  uniform  diameter,  101 

Breast-wheel,  225,  242 

efficiency  of,  250 
losses  of  effect  in,  248 
mechanical  effect  of,  247 

252 
speed  of,  246 

Broad-crested  weir,  58 

Bucket,  forms  of,  240,  250,  265 

Buckets,  number  of,  264 

Buff,  26 


Canal-lock,  time  of  emptying  and  fill- 
ing a,  29 

Capillary  tubes,  flow  in,  97 
Castel's    table    of   coefficients  of   dis- 
charge, 45 

Centrifugal  force,  effect  of>  255 
head  in  turbine,  298 
pumps,  307 

"       theory  of,  309 

"       vortex-chamber  in, 

309,  313 

Chamber,  whirlpool,  50 
Channel,  bottom  velocity  of  flow  in  a, 

154 

flow  in  an  open,  131 
form  of,  135,  136 
"         maximum  velocity  of  flow  in 

a,  150,  153 
mean  velocity  of  flow  in  a, 

151,  154 
mid-depth  velocity  of  flow  in 

a,  1ST 

"         steady  flow  in  a,  132 
"          surface  velocity  of  flow  in  a, 

150,  15-1 

value  of  yin  a,  144 
variation  of  velocity  in  a  sec- 
tion of  a,  148 
Channels,  differential  equation  of  flow 

in,  159 

examples  of,  162 
"          of   constant  section,  steady 

flow  in,  132 
"          of  varying  section,  flow  in, 

156 

surface  slope  in,  160,  161 
Chezy's  formula,  88 
Cock  in  cylindrical  pipe,  93 
Cocks,  loss  of  head  due  to,  93 
Coefficient  of  contraction,  22,  89 
"  discharge,   24 
"  friction,  73,  144 
"  "  resistance,  21 

333 


334 


INDEX. 


Coefficient  of  velocity,  20 
Combined-flow  turbines,  284 
Compressibility,  2 
Continuity,  2,  5 
Contraction,  imperfect,  22 

incomplete,  23 

loss  of  head  due  to  ab- 
rupt, 89 
Coulomb,  72 
Critical  velocity,  97 
Cunningham,  148 
Current-meters,  180 

Darcy,  72,  74,  75,  97,  148,  154 

gauge,   176,  178 
D'Aubuisson,  74 
Density,  2 

Downward-flow  turbine,  282 
Draught-tube,  theory  of,  301 
Dubuat,  154 

Elasticity  of  volume,  3,  4 
Elbows,  loss  of  head  due  to,  91 
Ellis,  106 
Energy  lost  in  shock,  32 

"       of  fall  of  water,  4 
"  jet  of  water,  27 

"         transmission  of,  84 
Enlargement  of  section,  loss  of  head 

due  to,  32,  91 
Equations,  general,  30 
Equivalent  uniform  main,  TOO 
Erosion  caused  by  watercourses,  136 
Examples,  60,  122,  170,  209,  315 
Exner,  183 
Eytelwein,  74,  134 

Floats,  sub-surface,  175 
"       surface,  175 
"       twin,  175 
Flow  from  vessel  in  motion,  16 

in  a  frictionless  pipe,  18 
"      in  aqueducts,  142 
"     influence    of    pipe's    inclination 

and  position  upon  the,  83 
"     in  pipes,  78 

"     in  pipe  of  uniform  section,  86 
"      "     "     of  varying  diameter,  98. 
Fluid  friction,  70 

"      motion,  I 
Fourneyron's  turbine,  281 
Francis,  176 

Friction,  coefficients  of,  70,  73,  74,  75 
in  pipes,  surface,  73,  97 
laws  of  fluid,  72 
Froude,  u,  13,  70,  76,  97 
Froude's  table  of  frictional  resistances, 
70 


Ganguillet,  147 
Gauge,  Darcy,  176,  178 
Gauges,  experiments  on,  148 
Gauging,  method  of,  173 
Gaugings  on  the  Ganges,  148 

"     "  Mississippi,  146 
General  equations,  30 
Gerstner's  formula,  229 
Graphical  representation  of  losses  of 

head,  94 
Grassi,  3 

Head,  2,  3 

Herschel,  184 

Hook  gauge,  Boy  den's,  173 

Humphreys,  148,  151 

Hurdy-gurdy,  279 

Hydraulic  gradient,  10 

mean  depth,  133 
"      radius,  80 
"          resistances,  20 
Hydraulics,  definition  of,  i 
Hydrodynamometer,  Perrodil's,  183 
Hydrometric  pendulum,  183 

Impact,  1 86 

on  a  curved  vane,  199 

on  a  surface  of  revolution,  192 

on  a  vane  with  borders,  195 

Inclination,  influence  of  pipe's,  83 

Injector,   12 

Inward-flow  turbine,  282 

Jackson,  148 
Jet,  energy  of,  27 
inversion  ot,  27 
momentum  of,  27 
propeller,  191 
reaction  wheel,  272 

efficiency  of,  274 
"       useful    effect    of, 
274 

Kutter,  147 

Laminar  motion,  2 

Lesbros,  27 

Limit  turbine,  283 

Loss  of  energy  in  shock,  32 

Loss  of  head  due  to  abrupt  change  of 

section,  89 

"      "       "    "  bends,  92 
"      *'       "     "  cocks,  93 
"       "       "    "  contraction  of  sec- 
tion, 89 

"    "  elbows,  91 
"       "     "  enlargement      of 
section,  91 


IAD  EX. 


335 


Loss   of   head   due   to   orifice  in   dia- 
phragm, 90 

"      "        "        "      "  sluices,  93 
"  "         »*••"«  valves,  93 

Losses  of  head,  graphical  representa- 
tion of,  93 

Magnus,  27 

Main  of  uniform  diameter,  branch,  101 

"     with  several  branches,  118 
Meters,  180 

"        inferential,  184 
"       piston,  184 
"       rotary,  184 
Meyer,  156 
Miner's  inch,  26 

Mississippi,  experiments  on,  148 
Mixed-flow  turbines,  284 
Motion,  fluid,  I 

"        in  plane  layers,  2 
"        in  stream-lines,  2 
"        laminar,  2 
"        permanent,  I 
"        steady,  I 
Motor  driven  by  water  flowing  along  a 

pipe,  107 
Mouthpiece,  Borda's,  34 

convergent,  44 
cylindrical,  39 
divergent,  42 
ring-nozzle,  37 

Navier,  149 
Notch,  54 

"      circular,  55 

"      rectangular,  54 

triangular,  56 
Nozzles,   104 

"         Ellis's  experiments  on,  106 

Open  channels,  131 

Orifice  fed  by  two  reservoirs,  115 

"      flow  through  an,  16 
:-•;*'      in  a  diaphragm,    loss  of   head 

due  to,  98 
in  a  thin  plate,  13 
"      in  vertical  plane  surfaces,  50 

with  a  sharp  edge,  14 
Orifices,  circular,  53 

large,  50 

"         rectangular,  50 
Outward-flow  turbine,  281 
Overshot  wheel,  225,  254 

"       arc   of   discharge    in, 

256 

bucket  angle  of,  262 
"  "       division  angle  in,  262 


Overshot  wheel,  effect   of    centrifugal 

force  in,  255 
"      effect   of   impact   on, 

270 
"       "    weight  on, 

268 
"       number  of  buckets  in, 

262,  264 

"       pitch-angle  in,  262 
"  "       speed  of.  254 

"       useful  effect    of,   268, 

271 

weight   of   water  on, 
256 

Parabolic  path  of  jet,  16 

Pelton  wheel,  280 

Permanent  regime,  i 

Piezometer,  9 

Pipe     connecting     three      reservoirs, 

branched,  in 
two  reservoirs,  86 
"    of  rectangular  section,  sluice  in, 

93 

"  uniform  section,  flow  in,  78 
"     "  varying  section,  18,  98 
Pitch-back  wheel,  272 
Pilot  tube,  176 
Plane  layers,  motion  in,  2 
Poiseuille,  96,  97 
Poncelet,  27,  227 
Poncelet's  wheel,  232 
Position,  influence  of  pipe's,  83 
Pressure-head,  4 
Prony,  74,  134 
Pumps,  centrifugal,   307 

"  "  theory  of,  309 

'•*  vortex  -  chamber 

in,  309,  313 

Radiating  current,  46 

Rayleigh,  Lord,  27 

Reaction,  190 

Reaction  wheel,  efficiency  of,  274 

"  "       jet,  272 

Regime,   permanent,  I 
Reservoirs,  Branched  pipe  connecting 

three,  in 
orifice  fed  by  two,  115 

"  pipe  connecting  two,  86 

Resistance  of  ships,  76 

"  to  flow,  law  of,  96 

Revy's  meter,  181 
Reynolds,  97 
Ring-nozzle,  37 
River-bends,  143 

Sagebien  wheels,  254 


336 


INDEX. 


Schiele  turbine,  284 
Ships,  resistance  of,  76 
Siphon,  108 

"       inverted,  109 
Slotte,  156 

Sluice  in  cylindrical  pipe,  93 
"      in  rectangular  pipe,  93 

loss  of  head  due  to  a,  93 
Sluices,  244 

Standing  wave,  165,  232 
Steady  flow  in   channels    of    constant 

section,  132 
Steady  motion,  i,  132 

"         in     pipe    of    uniform 

section,  78 
Stream-line,  2 
Suction-tube,  theory  of,  301 
Surface-floats,  175 
Surface-friction  in  pipes,  73 

•'       slope  in  channels,  160,  161 

Table  of  bottom  velocities,  155 

"  Castel's  results,  45 
"      "  coefficients  of  discharge,  24, 

25,  45 

"  friction,  73,  75 
"         "  velocity,  23 
"      "  density  of  water,  3 
"      "  discharge     through     nozzles, 

1 06 
"      "  elasticity  of  volume  of  water, 

4 
"      "  Ellis's  experiments  on  nozzles, 

106 
"      "  frictional  resistances,  70,  73, 

.74,  75 

"  maximum  velocities,  155 
"      "  values  of  f,  147 
"       "       "       "   "  Bazin's,  166 
"       ''  viscosity  of    water  and  mer- 
cury, 155 

Table  of  Weisbach's  values  of  Cv,  33 
Theory  of  suction  or  draught  tube,  301 

"        "  turbines,  284 
Thomson,  James,  50,  143 
Thomson's  turbine,  282 
Throttle  valve,  loss  of  head  due  to,  83 
Time  of  emptying  and  filling  a  canal 

lock,  29 

Torricelli's  theorem,  14 
Torricelli's  theorem  applied     to     the 
flow  through   a   frictionless  pipe  of 
gradually  changing  section,  18 
Transmission  of  energy   by  hydraulic 

pressure,  84 
Turbine,  axial-flow,  282 

"         centrifugal  head  in,  297 
"         combined,  284 


Turbine,  efficiency  of,  288,    291,  292, 

296,  297 
Fontaine's,  282 
impulse  or  Girard,  276 
inward-flow,  282 
Jouval,  282 
limit,  283 

losses  of  effect  in,  303 
mixed-flow,  284 
outward-flow,  281 
parallel-flow,   282 
practical  values  of  velocities 

in,  299 

radial  flow,   281 
Schiele,  284 
Scotch,  276 
theory  of,  284 
Thomson,  282 

useful  work  of,  292,  296,  297 
ventilated,  278 
vortex,  50 

Undershot  wheel,  225 

"       actual  delivery  in  ft. - 

Ibs.  of,  231 
depth    of   crown  of, 

238 
efficiency     of,     227, 

235,  239 
form    of   course    of, 

236, 

in  a  straight  race,  227 
"      losses  of  effect  with, 

228 

"       modifications   to    in- 
crease     efficiency 
of,  231 
"       number    of   buckets 

in,  238 

"       Poncelet's,  ^32 
"       Poncelet's  efficiency 

of,  235,  239 
useful  work  of,  228, 

235 

with  flat  vanes,  227 
Uniform  main,  equivalent,  100 
Unwin,  97 

Valve,  loss  of  head  due  to  a,  93 
Vane,  best  form  of,  199 

"      cup,  195 

Velocity,  bottom,  151,  154,  155 
"         critical,   97 

curve  in  a  channel,  152,  154, 
"         formulae,   150,  152,  154 
Bazin's,  152 
Boileau's,  153 
"         maximum,  151,   155 


INDEX. 


337 


Velocity,  mean,  151,  154 
"         mid-depth,   151 
"         of  flow,  286 
of  whirl,  286 
rod,  176 

"         surface,  150,  154 
"        variations  of,  119,  131,  148 
Velocities  in  turbines,   practical  values 

of,  299 

Vena  contracta,  14 
Ventilated  buckets,  272 
Venturi  water-meter,  13,  183 
Virtual  fall,  82 

"       slope,  10,  82,  84 
Viscosity,  96,  97,  119,  149 

Meyer's  formula  for,  156 
Slotte's         "          "156 
Vortex-chamber  in  centrifugal  pump, 

309,  313 

Vortex,  circular,  47 
"        compound,  50 

free,  47 

"        free-spiral,  48 
"        forced,  49 


Vortex,  motion,  47 

"        turbine,  50,  282 

Water-barometer,  5 
Water-meter,  13 
Weight  of  fresh  water,  2 

"        "  ice,  2 

"        "   salt  water,  2 
Weir,  54 

"      broad-crested,  58 

"      rectangular,  54 
Weisbach,  23,  26,  36,  76,  90,  91,  92,  93, 

145 
Wheel,  breast,  242 

hurdy-gurdy,  279 

jet  reaction,  272 

overshot,  254 

Pelton,  280 

pitch-back,  272 

Poncelet's,  232 

Sagebien,  254 

undershot,  225 
Whirlpool-chamber,  50 
Whirl,  velocity  of,  286 


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